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On Engel rings of exponent p-1 over GF (p) Chang, Bomshik 1959

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PROGRAMME OF THE  FINAL ORAL EXAMINATION FOR THE D E G R E E OF .  DOCTOR OF PHILOSOPHY of  -  BOMSHIK CHANG B. A., Seoul University, 1954 M. A., Seoul University, 1956  IN ROOM 207, ARTS BUILDING WEDNESDAY, APRIL 29, 1959 AT 11:30 A. M.  COMMITTEE IN CHARGE DEAN W. H. GAGE: Chairman G. R. R. S.  O. B. DAVIES P. DORE D. JAMES A. JENNINGS  F. B. D. A.  A. K A E M P F F E R N. MOYLS C. MURDOCH STROLL  External Examiner: PHILLIP HALL, M . A., F . R. S. Sadleirian Professor of Mathematics Cambridge University  ON ENGEL RINGS OF EXPONENT p-1 OVER GF(p)  ABSTRACT  It is well known that the;restricted Bumside problem for a prime exponent p can be rephrased in terms of the nilpotence of finitely generated .Engel ringsrover GF(p),with exponent p-1. r  We study these rings with the object of extending our knowledge of the Bumside groups. Let E^l be-the Lie-ring over-GF(p) generated by e^, -  ...,  e ^ , where the elements of E l are restricted by the Engel  condition [ fgP ] = 0 for all f , g fc E l . If l3 is the free Lie -1  ring over GF(p) generated by a j , . . . , a^, and if II is the ideal of L i generated by [ xyP" ] for all x, y t l 5 , then E ^ m / l / I I 1  We study E l by investigation~of-I9 in.lift; Let I ^ be the submodule of 1^ consisting of linear combinations of monomiai&inj a j , , I^ni,  ofj degree n , and let  n&q  . . . , n ) be the submodule of  1^  consisting of linear  combinations? of 'degree-nj* in a ^ , n2 in a2> . . . , ni + . . . + n^ =n.  in a ^ ,  The ranks of 1*1 and II ( n ^ , . . . , U q ) are  denoted respectively by i ^ and i l ( n j  t >  . . . , nq).  We prove-firsti that 111 issthe-module-'Spanned by all elements ofTtije.»formi and  [\v/f~^\ ,  and obtainjupp e r r bounds for i ^  j 1 ( n ^ , . . .„, nq)) which may be mosttconvenientty expres-  sed as coeffic.ients-of"certain-.formal powerrseries;  Further results are obtained by giving another set of eleo ments which spans This enables-us to-find upper bounds for i^(m, n) by, aniinductiVie method^ In^pjarticiUatrj we prove i^(p+r-n,n)-<  li  + -, K  where K is a polynomial in r of degree at most n-2. Using the above-formula, we prove that, if the Engel ringvE r'werej.nilpotent with-class. Cp,. then Cp/p.would not be bounded;. :  Finally, we give a new proof of the relation between the Bumside groups and the Engel rings by studying the free restricted Lie rings and Zassenhaus representation of the free groups. 1  PUBLICATIONS B. Chang, S. A. Jennings, and R. Ree, On certain pairs of matrices which generate free groups. Can. J . Math. 10 279-284 (1958).  GRADUATE STUDIES Field of Study: Abstract Algebra and Group Theory Modern Algebra  . . . ..  -  B. N. Moyls  Theory of Functions  V/. H. Simons  Functional Analysis  R. R. Christian  Other Studies: Symbolic  Logic  -  .  Electricity and Magnetism Theoretical  Mechanics  A.  Stroll  W. Opechowski F. A. Kaempffer  ON ENGEL RINGS OF EXPONENT p - 1  OVER G F ( p )  by BOMSHIK M. A., S e o u l  CHANG  University,  1956  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  t h e Department of MATHEMATICS  We  accept  standard  this  thesis  required  degree of Doctor  THE  as c o n f o r m i n g  from  candidates  to the  f o r the  of P h i l o s o p h y :  UNIVERSITY OF BRITISH COLUMBIA April,  1959  - i i-  ABSTRACT It problem of  i s well-known  f o r a prime  the n i l p o t e n c e  exponent  w i t h the exponent  these  rings  the  with  Burnside Let  the  £  f,  I  be  g  Cj ^  generated  we  o f e x t e n d i n g our  by  We  [xy  n,  module of  degree + n  q  free  generated  by  the  are  L i e r i n g over  a , and i f J  study  denoted  g  let  i s the  "J ^  i n a,,  ^ ( n ^ n j , • . .,n^)  The  ranks  spanned by  of  J  7^ n  by  first  a l l elements  upper  of  "J ^  consisting a  o r  ....  2 the  sub-  l i n e a r combinations  and  i ^ and that  J^  q  (n , , n  of  2'  0  J °' i s t h e module [xy  ],  and  i (n^,n^,•••,n^) in q  ...  . . . . , n ) are ' q  i^(n^,n^,•••,n ).  of t h e form  bounds f o r i ^ and  1'  a g  i n a^,..., n^ i n a , n^+  respectively prove  be  ideal  > then  investigation  o f monomials  of  GF(p)  y £  submodule of  consisting  We  obtain  and  n^ i n a^,  =n.  study  ] = 0 for a l l  [fg  "*"] f o r a l l x,  P  l i n e a r combinations degree  over  knowledge  GF(p)  1' of  i n terms  Engel rings  In t h i s t h e s i s  condition  i s the  L e t 'J ^ be of  rephrased  L i e r i n g over  by a^, a^,...,  XJ^IJ ^.  S  the  the E n g e l  generated  q  be  e , where t h e e l e m e n t s o f  If  e  generated of  by  r e s t r i c t e d Burnside  groups.  by e^, e^,..., restricted  p-1.  object  the  p can  of f i n i t e l y  GF(p)  of  that  - i i i-  terms  of w e l l d e f i n e d i n t e g e r s j  and  j ( n , , n . , . . ,,n ) ± <s g  n w h i c h may of  be most  certain  conveniently expressed  formal  power  Further  as  coefficients  series.  results  are  o b t a i n e d by  giving  another  2  set  of  elements which  spans  •upper bounds f o r i (m,n) particular,  we  an  . This enables  us  to  i n d u c t i v e method.  In  prove i  2  r"" (p + r - n , n ) < ~. rr (n-i;| 1  where K i s a p o l y n o m i a l Using  by  'J  the  +  K,  i n r of d e g r e e  above f o r m u l a ,  we  n-2.  at most prove  that,  2  if  the  then  Engel  ring  £  c /p would not  were n i l p o t e n t be  with  class  c ,  bounded.  P We about the  the  Engel  rings  and  groups .  give  another  proof  relation  between t h e  rings  s t u d y i n g the  by  Zassenhaus'  of w e l l - k n o w n Burnside free  fact  groups  and  restricted  r e p r e s e n t a t i o n of the  free  Lie  find  Faculty of Graduate Studies  PROGRAMME OF THE  FINAL ORAL EXAMINATION FOR THE D E G R E E OF  DOCTOR OF PHILOSOPHY of  BOMSHIK CHANG B. A., Seoul University, 1954 M. A., Seoul University, 1956 IN ROOM 207, ARTS BUILDING WEDNESDAY, APRIL 29, 1959 AT 11:30 A. M.  COMMITTEE IN CHARGE DEAN W. H. GAGE: Chairman G. R. R. S.  O. P. D. A.  B. DAVIES DORE JAMES JENNINGS  F. B. D. A.  A. K A E M P F F E R N. MOYLS C. MURDOCH STROLL  External Examiner: PHILLIP HALL, M. A., F . R. S. Sadleirian Professor of Mathematics Cambridge University  ON ENGEL RINGS OF EXPONENT p-1 OVER GF(p)  ABSTRACT  It is well known that the restricted Bumside problem for a prime exponent p can be rephrased in terms of the nilpotence of finitely generated Engel, rings over GF(p) with exponent p-1. We study these rings with the object of extending our knowledge-; of the Bumside groups. Let E1 be the Lie ring over GF(p) generated by e^, . . . , e^, where the elements of E l are restricted by the Engel condition [ f g P ] = 0 for all f, g £ E l . If L i is the free Lie ring over GF(p) generated by a^, . . . , aq, and if I i is the ideal of L i generated by [ xyP" ]' for all x, y C L , then E 1 ~ L 1 / H We study E l by investigation o f 111 in. L I . -1  1  q  Let 1^ be the submodule of i i consisting of linear combinations of monomials.in- a^, . . . . , q i - ° t degree n, and let H ( n i , . . . , n^) be the submodule of I i consisting of linear combinations of'degreeing in a-^ , 112 in a£ , . . . , in a^, ni + . . . + n„ = n. The ranks of I i and II (n. , . . . , n ) are 4 Q n i» » 4 denoted respectively by r* and i i (nj', . . . , n^). a  1  n  We prove, first that Ii'. is-the:module-spanned by all elements of.the form. [ xyP ^] , and obtain upper bounds for i and j1(n^, . . . . , nq) which may be most:conveniently expressed as coefficients of certain formal powerr series. -  q  Further results are obtained'by giving another set of elements which spans I ; This enables.us.to find.upper bounds for i^(m, n) by an.inductive;method.. In particular, we prove ? -ii.-1 i^(p+r-n,n)<- l ! + K', (n--l)j where K is a polynomial in r of degree at most n-2. Using the above formula, we prove that, if the Engel 2 ring E - were nilpotent with class Cp., then C p / p would not be bounded. Finally, we give a new proof of the relation between the Bumside groups and the Engel rings by studying the free restricted Lie rings and Zassenhaus' representation of the free groups.  PUBLICATIONS B. Chang, S. A. Jennings, and R. Ree, On certain pairs of matrices which generate free groups. Can. J . Math. 10 279-284 (1958).  GRADUATE STUDIES Field of Study: Abstract Algebra and Group Theory Modern Algebra  B. N. Moyls  Theory of Functions  V/. H. Simons  Functional Analysis  R. R. Christian  Other Studies: Symbolic  Logic  Electricity and Magnetism Theoretical Mechanics  _.. _ .  A. Stroll W. Opechowski  - F. A. Kaempffer  In p r e s e n t i n g the  this  r e q u i r e m e n t s f o r an  thesis in partial  advanced degree at the  of B r i t i s h Columbia, I agree t h a t it  freely  agree t h a t for  the  a v a i l a b l e f o r r e f e r e n c e and permission for extensive  s c h o l a r l y p u r p o s e s may  D e p a r t m e n t o r by  be  gain  shall  a l l o w e d w i t h o u t my  Department The U n i v e r s i t y o f B r i t i s h V a n c o u v e r $, C a n a d a . Date  study.  I  Columbia,  make  further  copying of t h i s  his representatives.  c o p y i n g or p u b l i c a t i o n of t h i s be  shall  the  of  University  Library  g r a n t e d , by  that  not  fulfilment  Head o f  thesis my  I t i s understood  thesis  for  written  financial  permission.  - i v-  TABLE OF CONTENTS  Introduction  . . . .  1.  Free  L i e ring  2.  Some  identities  3.  The E n g e l  4.  The i d e a l  5.  Lower bounds  6.  Burnside's  7.  U n s e t t l e d problems  Bibliography  fj* o v e r  ideal J  1  . . . . . .  in J  CK  6  GF(p) . . . . . . . . . .  .  9 14  q  21  2  f o r the c l a s s  groups  of  . . . . .  43  . . . . . . . .  48  . . . . . .  66 68  -1-  Introduction. integers,  or a f i e l d .  <j)-module  i n which  pair  of e l e m e n t s  Let  ^  be e i t h e r  A Lie ring  over  x and y of £  is a  are  i s defined,  postulates  (0.1)  A.(xy) = (A.x)y = x ( \ y )  and  (0.3)  fulfilled:  + z) = xy  + xz  (Anticommutative  law)  a consequence  (x, y, z £  o f t h e above p o s t u l a t e s ,  (Jacobins  A subring,  i f xy £ JL'  (0.3)  a two-sided  (0.5)  JLV) .  have  a Lie ideal,  JJ,  also.  Ji,),  x, y o f JL,  ideal  f  if  JL . ( i f xy £ Xj  and y E  Thus any  z £  a Lie  of e l e m e n t s  o r s i m p l y an  x £ fj  elements  (x, y,  JL* i s c a l l e d  of  f o r any p a i r  yx £ £j  T  Lie ideal is  ideal.)  For central  H  $-submodule  xy £ JL' f o r any  X,) ,  identity)  -C' i s c a l l e d  by  we  (x, y £  + (yz)x + (zx)y = 0  (xy)z  then  (x £  xy = -yx  1  (0.4)  and  L),  (A. £ <3?, x, y e  xx = 0  (0.3)  the  (x + y ) z = xz + y z , x(y  As  of  t h e L i e p r o d u c t xy f o r e a c h o r d e r e d  following  (0.2)  the r i n g  any  Lie ring  £^ we  may  form the  lower  series:  1,  -  ^ 2  1(  2 >  2  1^2.  ... ,  where  (0.6)  +  A Lie ring  is said  (o.7)  f  ^^Xi)  ( i  t o be n i l p o t e n t  = 0, (n + l )  l l  2  '  of c l a s s  * 0. (n)  =  "' n. i f  )  '  -2-  We  may  L  (0.8)  also  =  JL  form  the  ^  ( 1 )  derived 2  ^  (  3  series  2  )  ...  JL- :  of ,  where  (o.9) and  tf  i  +  1  a Lie ring  =  )  JL  «L is said  (o.io) for  X,  some  said  t o be  to  elements  y o f JL  is called Engel  the re e x i s t s  depend on  (en)  an  x and  Engel  condition  y,  ring,  i f for  a positive  such  n  or  any  integer  n,  that  ^  ((...((xyirrrTTTyjy = o.  particular,  called  an  satisfy  i f n i s independent  Engel  ring  of  the nth Engel  if L  that  i f «L  satisfies  condition  shown t h a t ring  or  rated  Engel  still  open.  has  a solvable  The  of  JL  o r X-  (Engel's finite  nth Engel  and  ring  characteristic  to  over  says  a f i e l d <£>  then  L  is  i f JL  satisfies  i f JL  s at i s f i e s t he  Higgins  of exponent  [8]  the  has  n over  a  p r i m e t o n,' i s  t o whether  of e x p o n e n t  theorem)  rank  is nilpotent.  as  is  is said  condition  shown t h a t  Engel  question  ring  n,  y t h e n JL  ,  f o r subrings, then  a field  nilpotent.  the  [30]  Zorn  maximum c o n d i t i o n Engel  theorem  <J>-module of  is a  nilpotent.  exponent  of x,  condition.  A well-known  and  solvable i f  - o  ( n )  JL  s a t i s f y the  x,  w h i c h may  In  ),  n. A' L i e r i n g  is  (i = l . 2, . . .  (i)  a finitely  n is nilpotent  gene-  i s , however  /  -3-  The a special rings have  class  of E n g e l  over GF(p), important  celebrated  given  of t h i s t h e s i s  rings,  of exponent  applications  namely,  p-1.  somewhat  g r o u p we  product  a ring  and t h e commutator  (Lazard [13]/  indicate  i n such  this  Magnus [ 1 8 ] ) .  If the given  the associated  Lie ring  (Higman [ 9 ] »  condition  element' satisfies  Sanov  Hence t h e n i l p o t e n c e  of t h e " f r e e " E n g e l  q generators  the p r o p o s i t i o n  implies  corres-  of L i e elements  g of the group) then Engel  that  of group elements  p ( i . e . g^ = 1 f o r every  (p-l)st  To a  a way  group i s of exponent  the  type  with the  as f o l l o w s .  pond t o t h e sum and t h e L i e p r o d u c t respectively  Engel  of t h i s  i n connection  intuitively,  associate  i s t o study  free  Rings  B u r n s i d e p r o b l e m . We may  connection,  the  main p u r p o s e  R  [26]).  ring  with  for p = 5  Indeed, t h e s o l u t i o n  of the problem  [9]/  Kostrikin  [12])  i s b a s e d on t h i s  fact.  for  i = 1,  ...  2p-2,  of the f a c t o r  2,  C J (j^+i)  rings orders #3  n  of the f a c t o r  denote and  a  r  e  e c  ?  review  u a  1,  some c o n c e p t s  l  ^3 ^ ^ ^ i + i '  w  ^  to the e  r  e  we  a n c  s e r i e s of  introduce  concerning ring,  some  notations  the free which  Lie ring  are used i n  sections. In  section  2,  we  *  £) .  o v e r G F ( p ) and i t s d e r i v a t i o n later  respectively  (Higman  Moreover,  of t h e lower c e n t r a l  the Burnside group section  the orders  groups  the n t h term  In and  ,  6).  (cf. Section  consider  the d e r i v a t i o n  -4-  ring  and i n t r o d u c e  {  l  X  X  [10]  Jacobson's  2  . .. 1 )  1 2  where t h e X^ a r e n o n c o m m u t a t i v e minates i^,  and t h e summation  i^,..*,  o f 1,  these  symmetric  Jacobson.  I t i s well-known  is  a L i e element  [ll], the  Zassenhaus  symmetric  sum  a L i e element -  X  - Y  2  P  over  a l l permutations  X . We  - X  of which  P  properties  were o b t a i n e d by  = A(X,Y)  (cf. Section  1.2)  [28]),  implies  which  ... X ^ / l 1  {X^  some  that,  - Y  P  list  (mod p)  i n X and Y ( j a c o b s o n that,  i n general,  ... l } (mod p) i s  i n X^, X^,..., X^. F o r example,  (X + Y)  = XY + YX = XY - YX = [ X Y ] (mod 2 ) . We  2  indeter-  2,..., n i v h i c h may be c a l l e d t h e  sums most  (X + Y )  i  but a s s o c i a t i v e  i s taken  sum o f X^, X 2 , . . . ,  symmetric of  i  notation:  prove  that,  [x (x ... x 1  /1 1 . . .  2  =Z[...[[x x  1]]  1  where a g a i n t h e summation ig/**"/  sum  {X^ X^ ... X /1 1  the  free  i p o f 2,  ... l } . T h i s  r o l e throughout  this  In  define  s e c t i o n 3#  Lie ring  w  ©  a  n  fact  plays  a most  thesis. the Engel  d prove  that  of t h e form the Engel  module g e n e r a t e d by a l l e l e m e n t s the  a l l permutations  s e t of a l l l i n e a r  *J o f  ideal  X , o v e r G F ( p ) t o be t h e i d e a l  g e n e r a t e d by a l l e l e m e n t s x, y £ JL ,  over  ] X . ]...X. ] , 2 3 p  3,..., p, i s a c t u a l l y t h e symmetric  i-^,  important  i s taken  1  of £,  ((...(xy)...)y)y,  ideal  i s the  ((...(xy)...)y)y,  combinations  of such  elements  i.e.  -5-  (Theorem 3 - l ) - U s i n g for  the  lower  ranks  this  of t h e module  bounds f o r t h e  L^) /E) i  f o r i = 1,  i+ 1  ranks  bounds  . , which w i l l +l  give  of t h e  abelian factor  groups  2p-2.  4,  of t h e  o b t a i n upper  J ./T i i  2,  In s e c t i o n consideration  t h e o r e m we  we  restrict  Engel  ideal  o u r s e l v e s to  'J,  of t h e  a  free  Lie  2 ring  £,  with  special this of  type  X/  such  an  of b a s i s  t h a t any  the  n i n the  other.  In ranks  for  set  i n one  as  (Theorem  4-2).  obtain ranks  of a l l  generator  !  u s i n g the  formula  of E n g e l  generator  In between t h e We  ring  cannot  of p i n g e n e r a l  this  5/  written  consisting  degree m  elements  u p p e r bounds f o r t h e  ^J(m,n) o b t a i n e d i n s e c t i o n 4 t o g e t h e r w i t h  nilpotence  with  be  shall  v  section  well-known W i t t ' s  one  may  Using of  u p p e r bounds of t h e  ^J(m,n) of  homogeneous e l e m e n t s w i t h and  j  a  X_?«  o b t a i n a subset 4 2  t h e o r e m we  f o r the  submodules  first,  of e l e m e n t s of t h i s  of t h i s  expression  choose,  (a n o r m a l b a s i s ) f o r  element: of  combination  application  another of  g e n e r a t o r s . We  s e t of b a s i s e l e m e n t s we 2  a linear As  two  (Theorem  section preceding list  thesis  be  6,  we of  prove  t h a t the  exponent  expressed  w i t h more  of than  as a l i n e a r f u n c t i o n  5«l)« we  d i s c u s s the  results  and  the  some open q u e s t i o n i n the  p-1  class  last  section  connection Burnside  in 7«  groups.  connection  -6-  1. follows  we  Free  denote  g generators We  a^,  s h a l l denote  The  by  will  be  called a  and  will  of  the  a2,«««,  An  , or  of  by  i d e n t i t y enables  in the g e n e r a t o r s  ^  Ju.  letters.  1^ w i l l  be  (...((x.x»)x„)...)x 1 2 3 n  product A normal  will  product  be w r i t t e n  as  xzy.  us t o w r i t e e v e r y of  element  ( r i g h t ) normal  a^, a^,...,  submodule ;.of t3  be  a^.  consisting  denoted  . Similarly,  of  products  of  a l l linear  o f homogeneous monomials o f d e g r e e  the g e n e r a t o r s w i l l by  form  with  s i m p l y by  in  x-^x^x^. . . x^.  as a l i n e a r c o m b i n a t i o n  arises,  y  of t h e  x ( y z ) = xyz -  combinations  GF(p)  have  (1.1)  The  In what  small Latin  x and  ( r i g h t ) normal by  GF(p).  L i e r i n g over  xy...yz...z...u...u  xy^z"' . . .u^. We  This  jtl_5 o v e r  a ^ by  element  denoted  form  free  of elements  denoted  be  the  elements  L i e product xy.  Lie ring  by  Xj^i  o  r  if  £.(  n  n in confusion  o  ) will  denote  t h e submodule o f  c o n s i s t i n g of a l l l i n e a r c o m b i n a t i o n s n of homogeneous m o n o m i a l s i n e a c h g e n e r a t o r w i t h d e g r e e n.. i n a,, n„ i n a „ , . . . , and n i n a , where n + n~ + 1 1 2 2 ' ' q q' 1 2 n  ...  + n  = n.  q  The  £__(n.., n , . . . , n 9  respectively.  ranks ) will  be  denoted  by  E x p r e s s i o n s f o r f ^ and  have been g i v e n by W i t t  (1.2)  of the modules  f* -  i  [27], v i z  Z^U)q d | rv.  n / d  ,  1** ^n f  q  and  and  f ( n , , n ,...,n  f ( n ^ , n2»...,  p  n  )  )  -7-  M-(d) (1.3)  f(n  n  l f  n ) -  2  (n  + n  x  where U-(d) i s t h e M o e b i u s  1.2. An  inner  itself is  The f r e e  | ... _ S |  = n),  g  function.  associative  r i n g ( X ^ over GF(p).  a mapping  of  o f jL, i n t o .£_,  T h r o u g h o u t , we d e n o t e  by t h e same c a p i t a l l e t t e r  the  f i x e d element  for  example  of X - d e f i n i n g  Y_^ w i l l  The i n n e r  mapping  x —> x(yz) w i l l  derivation  the inner  defined  derivation;  x —> xy^ f o r a l l  by y z , i . e . t h e  be d e n o t e d by [ Y Z ] . By ( l . l )  have  (1.2.1)  [ Y Z ] = YZ - ZY. Again,  for brevity,  we w i l l  write  (1.2.2)  [...[[X X ]X ]...X ] = [X X X ...X ],  (1.2.3)  [XY...YZ...Z...U...U] = [XY Z ...U ] .  1  The by t h e i n n e r will  of  2  3  associative derivations  n  1  2  3  n  r i n g over GF(p) generated A^: x — » x a , ,  i =  1, 2,..., q,  be d e n o t e d by 0 ( 5 o r (X. I t i s known t h a t  a free The  inner  as i s u s e d f o r  d e n o t e t h e mapping  x £ JL-  we  *-1  by Y: x — * xy f o r a l l x £ JL , where y  a f i x e d element  derivations  | _2  + ... + n  Y of X/ i s  derivation  defined  2  c*K _1  d  r i n g with generators  e l e m e n t s o f (X w h i c h  A  J f  OL i s  A^,..., A q ( W i t t  are themselves inner  [27]).  derivations  J L form a L i e r i n g i f t h e L i e p r o d u c t o f X and Y,  -8-  where X and Y a r e i n n e r [XY] or  = XY - YX. T h i s  simply  xy —> We  by £  Lie ring  be d e n o t e d by ^3  [ X Y ] i s an i s o m o r p h i c mapping o f  shall  call  the elements  L i e elements  as a r e s t r i c t e d element called  A^,  L i e algebra  not  an element  shall  will  = A^A^ -  ([lO],[ll]) X e  so t h a t  will  an  not be  mapping o f «£_, d e f i n e d  be d e n o t e d by E. Note  we w i l l  consider  x = xE, xy = xY, =  x[YZ],  xyz = xYZ,  ect.  by x —>  that  E is  i n the  E adjoined  use t h e f o l l o w i n g c o n v e n i e n t  x(yz)  A^A^, . . .  consider  o f (X, b u t , f o r c o n v e n i e n c e  sections  ([27] ) .  o f OC.  identity  a l l x £ X_»  onto ^  o f ( X . We p r e f e r n o t t o  a L i e element  following  q  the L i e elements of  [A^A^]  p  for  (1.2.4)  o f £f  of C L of the form X ,  The  We  will  i s d e f i n e d by  . The m a p p i n g x — * X, y — * Y and  jQ[: f o r example A ^ . . . , are  derivations,  to  natations:  x  -9-  2. we  list  which  certain  will  proofs  Some i d e n t i t i e s identities  be u s e d  Let  X^,  IX.. X„ ... X \ <  l i  <s  1  i  1  2  where than  n ... i  i^,  sections.  [10] n o t a t i o n  We  the  n  1 2  I  l  i„  are non-negative  2  p, w i l l  denote  particular, | I  X  ( 1  t h e sum  i  ... 1  2  give  }) n  integers  ][  X  2  smaller i ^ of  order.  . . . X /1 r  X^,  In 1 ... l } )  J X.  X.  ...X,  , where  are taken over a l l p e r m u t a t i o n s of  n  n•  + ... + i by X^,  n  the  Y^  and  e a c h of t h e  (2.1)  ( o r {X  of n l terms  Consider + i  i  of a l l p r o d u c t s o f  ... X )  2  1  d e n o t e s t h e sum  2, • t • |  ...  2  #  1,  not  j  i , . . . ,  i_,..., 2  shall  (X  symbol  i„ of X. ..., i of X i n every p o s s i b l e 2 2 n n  i,, 1  of  X, Y be e l e m e n t s of GC-  {X, X„ ... X / i .  (or  section  formulas.  X^,..., X^,  Using Jacobson's  In t h i s  among t h e e l e m e n t s  in later  f o r well-known  in a .  ( X) 1  ^1  ) r  [Y. Y„ 1 2 a  n  d  ... Y / l 1 ... l } , (m = i . m 1  replace  each of the next last  each of the f i r s t i  2  of t h e Y^  i  by  o f t h e Y. by X . Then we n k n • a a X- X» X— a • a X a a • X ] 1 Z 2 n nl • • • 1 1 a a • 1 •a a 1 a n a l / m  m  i ^ of X , 2  have  m  = V - V - V - F l 2 ••• n { X  Another  rather  X  s i m p l e but u s e f u l  identity  is  -10(2.2)  | X  X  x  ... X  2  n  I i , i ... i 1 2 n = X l) l n} l i 1-1 2 i . . . . i n J 0  v  X  X  X  +  X  J  2f l 2 ••• M l i 1. 2i ~ - l . . . in / X  + X IX. X, ...  1 2 i , i„ . . . X— ... 1X2 X. + 2 n, 1 i ... i j n)  — \ X,  i  1  [i^i  2  + ..•.+  X  n i -1 \n X. X~ ... X ) IX nl 2 j 1 2 nl C i i - i ... i J 0  n  f x . X. ...  X  1  X  2  n  \X  ^ 1 2 n \ n li i i -lj • The following two identities are well-known([10]) x  (2.3)  ( X  X  x  (1 "  (X  1  +  2( !  j  ... 1  +  X  X  +  n  ... X n |  2  1  2  2  +  X 2  " -  "2>( 1  +  +  X  •  V 2  +  )  n  +  X  ~ '••  2  +  +  "  V l ^  +  (-l) ~^ " • n  x  Here X \ (X 1 ., + 2X_ +...+ nX- i , ) denotes the sum of the ( . ) terms ( X + X +...+ X ) where j . , n-i 3 J n-i n  n  n  i  Ji  i  1  2  2  ±  runs over a l l combinations of 1, 2 , . . . , n taken n - i at a time.  | x + x +...+ x x t - Z U + x +...+ x .  (2.4)  x  2  I  n  + . . . + (-D  1j  n  n _ ]  j x^  X  \ 1 1 where  2  C  2  n  (  5 fx X \ x  C  =  X  n  (n  1)  ... X^ X 1  ... 1  1 ) ,  has the same significance as in (2.3). (The  j . n-i  -11-  identities  (2.3) In  are  the  assumed t o  identities  and rest  be  are  (2.4) of  hold  i n any  associative  this section  e l e m e n t s of  (3.  a l s o well-known,  X,  The  X,,  Y,  ring.)  Y,  I  I  following  two  ( c f . (1.2.3) f o r  the  not at i o n s . )  (2.5)  [XY ]  = XY  (2.6)  [XY  ] =  P  -  P  Y X. P  p-i  (2.7)  P - 1  so  p-i-1 _  = iX I1  of  XIX. X« ... X J 1 2 p-1  \  clarity  that  I  A generalization  ^1 (For  '2,Y XY  the  ...1  JJ  stress  again  1 we  le.fi.t-v.hand s i d e  (2.6)  Y I P-1 J -  i s the  — IX X. X— ' 12  (1 1 our  1  identity:  ... X  } p-1 \  ...1  ). (1.2.2),  convention  denot 1 1 X  where t h e summation l ' '"*"' p - l ° Proof of i  i  i  f  2  2  p - 1  n  n  X  1  •x •  1  ]  X ^ ^ P -  2  1  • (-D^'Zcxxr ] 1  c  1  = I X X- + x + . . . « - x . l 0  J  1  2  -  £ | X c  P-1 J  p-1  11  l l  X  x  1  2  1  p-l  i s taken over a l l permutations ' **"' "^ (2.7). By (2.3), we have  XIX X~ ... X I 12 p-1 1 ... 1 = [ x ( x + x 2 + .+ x p _ x ) p  -Z[x(x  l  1  • X  2  +  ...  p-1  +  X  1 )  ]  -12-  ( " D ^ Z j X  X  1  1  This well-known  X  x  x  " l  C  X  X l  ... X _ \ p  1  1  i d e n t i t y gives  formula  another  (Zassenhaus  (2.8)  = X  P  + Y  P  = X  P  proof  of t h e  +  P  /\(X,Y),  o f an a s s o c i a t i v e r i n g o v e r a  i n X and Y.  (X + Y )  (2.4)).  [28]):  of c h a r a c t e r i s t i c p > 0, and  L i e products  (by  ... 1  where X, Y a r e e l e m e n t s  of  (2.6))  p-1  2  (X + Y )  field  (by  p  + Y  /\(X,Y) i s a sum  Indeed, + 21) X ~~'(n  P  Y \ p-nj  n  and P-!  A  P-I  (X,Y) = ZJ h  s  IX ,  Y  l n  (X  + X  +...+  2  X )  Y 11  " - ' l ln-1  p-nj  A s i m i l a r argument e s t a b l i s h e s (2.9)  = Z - f X| X  \  p-nJJ  .  the i d e n t i t y :  P  k  = X + X +...+ x + /\(x^, x , . . . , Xj__) , P  P  p  2  where  A (x^, X , . . . , 2  x^) - 21 \ x^ x 1  2  . . . xx,  n  v  =  /.r x,iX J  L  n  1  ( i]_  x~ ••« x , 2  *2  "** * j ^  The f o l l o w i n g ring  (2.10)  JL^ a r e i m p l i e d x  j  k  *" "  -  * 0,  x  ±  important by x  1  + i  2  +...+  i  k  = p).  r e l a t i o n s i n the f r e e  (2.7):  i | 2 *** i ""* p v. X ••• X •• • X x  ... Xj ... x^  3  (i  Lie  2  =  X  il 2 X  **" I1  X  l p] •• • X i t X  i  1 J  i  -13(2.11)  ytx  1  I (Again  ... x l  2  1  1  ... 1 I  ... 1  2  {1  ... x i y  ... 1 J .  ... 1 )  t h e summation  i s taken  j 2'"••» J i ' • " " '  °f ^*  d e n o t e s ^> x, x-  ...x^ y w i t h  as  = -x ,x  x . i x„ ... x. ... x | d e n o t e s ^ x . x , ...x, ...  I 1 where  x  1  over  a l l permutations p  ,  X , ..• ,  and Xjj  x  2  I1  t h e same c o n v e n t i o n  on  above.) Proof y x  1  x  1  1  f  x  x  \ = y\ i x  P  . 1 J  (1  2 ••• p l 1 ... 1 ) X  X  X  = ~x jx 1  Again  (2.12)  .  2  in  , u s i n g the f o l l o w i n g  identity  Y-,Y«...Y Z — ZY-.Y.-....Y  12 = [ •  (2.13)  •  2  ^ lJ 2  =  we can  11).  (2  of  Y l  12  n  Z]Y ...Y 2  n  + Y  l L  n  Y Z]...Y 2  n  +  . • •  y Y ...[Y z], 1  2  n  establish  J 1 2 ••• n ] " j l 2 ••• n ] ll 1 . . . l j ( l 1 ... 1 ) Y  Y  [  Y l  Y  Z] Y  1 • '  1 +  2  Z  ... Y  Z  Y  Y  l  +  ... 1 )  Y  (Y  I1  x  [Y Z] . 2  1  i l 2 ••• £ n H 1 1 ... 1 ) . Y  Y  Y  Z  Y n  1  J  -14-  3.  The  g e n e r a t e d by called the  a l l elements  the E n g e l i d e a l  Engel i d e a l  only  fixed  ^  The  q  of  i s spanned  i f every elements  as" a ( f i n i t e ) l i n e a r  be  or simply  dealing  be d e n o t e d by  confusion,  by  p-1  s h a l l be  and w i l l  in  will  o f exponent  Xj"*/ s i n c e we p-1,  Lie ideal  o f t h e form xy  order to avoid  a module  a ring  of  exponent In  that  Engel i d e a l  we  shall  be  c o m b i n a t i o n of e l e m e n t s  J.  or  q  say  a s u b s e t of e l e m e n t s  o f t h e module can  with  of  written of the  subset,; L e t fa- d e n o t e all  elements  the  notations  of the form Y  g e n e r a t e d by J  or the  defined  xZD,  x £ X^,  By 1  Y  shall  of  P  2  ...  Z £  element  (2.3)  if Y  J  spanned , and  of  (t .  prove the f o l l o w i n g  theorem.  Theorem 3•1•  ideal  The  by  a l l elements  Proof.  C o n s i d e r an  P _ 1  z  p  = -z  the  E $,  Z £ fH  t  of  identity  1  then  Using t h i s  J  i s the  fact,  p—  1  xy  element  "'"z £ *J.  xy  -z[XY  P  P _ 1  ]  we  submodule  o f t h e form  = - (xy - ) = Z  £^  a l l elements  D is either  2  l  xy  of  OC.  Y^_ /1 1 ... 1} £  oC spanned  by  Y ,..., Y ^  l f  x £ t^,  xZ,  by  using  i s the i d e a l  of t h e form  i s t h e submodule of form  0( spanned  Y £ 3" • Then  P  (1.2.4),  in  a l l elements  mapping E o r an  [Y  t h e submodule of  Then  -15-  = -zXY  -zY|X \l  P _ 1  = -(zx)Y ~ P  -  1  Y  \  P-2l  (zy)iX  Y  \l But  Y  e (t and  P _ 1  )X  U  Y  \ e  (2.2))  (by }  .  p-2)  , so t h a t  xy  p - 1  z  is  P-2J o— 1  a linear u,  combination  only  the  the p r o o f  fact  that  characteristic  of t h i s  form  over  p and  generated  a field by  the  exponent  shows t h a t  of  a l l elements  xy  t r u e i n g e n e r a l . (A c o u n t e r  constructed.)  over  uv  ,  have  used  a field  of E n g e l  characteristic p— 1 a l l e l e m e n t s xy , x,  module s p a n n e d by not  t h e o r e m , we  is a Lie ring  o t h e r words, t h e p r o o f  ring  is  of the  v £ Xj • In  In  of elements  if p then *p y £  ideal i s any the  is  is  p-1.  Lie  ideal  i s the  . However, example  of  this easily  -16-  3.2. Theorem 3«1 is  The  upper  and f o r m u l a (2.3)  the module spanned  x,|x„  ... x  1/2  p  ... 1  11  form  by  1)2  x^,  . ...  1  .  x p  .  o f t h e form  l J  lf  ... 1  element  m  of  y  a  x,ix„  ^  n  2'""''  J  ^ J ( n ^ , n^, *••,  of  i(n  1'  v  It exists  be  f  ...  2  s e l e c t e d from  p  n  q) •  be  n )  2  g  r a n k s o f t h e modules  n ) will  ( c f . M.  be d e n o t e d by give  upper  element  a ,..., 2  and  a ,..., <c  a  X^.  a^ o f  l e t x^,  elements  q  of  i s a homogeneous  of the degrees 0  bounds  Hall [5]) a basis  among t h e s e b a s i s t h e sum  i ^ and  follows,  such a b a s i s ,  i n t h e g e n e r a t o r s a,, 1  Denote  1  and J ^ , n  n  every basis  a l l o w e d ) such t h a t x  x. i s  1  1  i n t h e g e n e r a t o r s a^,  u^, u , . . .  by a l l  pi  n „ , . . . , n ) as 2' q  i s known  such that  expression Let  1  . . . l j  spanned  n^,•••, n ) r e s p e c t i v e l y . We  i ^ and n  p V  £/.  ^ ( ^ /  i(n^,  p, i . e . ,  1  ...1  e  J  ... x ) where e a c h  1/2  = JJ n  n  ... 1  I  q  ^ and  1  (1  j  Let Jl = J nV g  J  ideal  of t h e .form  11 2  U basis  the  1, 2 , . . . ,  i =  p(  1  ... x!  elements  a l l elements  I  1  ...  show t h a t  J  ...  li  n  0  1  + x^jXg  j ( n ^ , n,-,, . . ., g ) •  ) , x. £f. . Since x-,fX ... x 1 i s a l i n e a r 1. *° II 2 p\  i n e a c h component  \l  bounds j ^ and  x  2 >  x  p  (repetitions  o f x^,  i s equal to  by j ^ t h e number o f ways i n w h i c h  ...,  x ,..., 2  n.  such a s e l e c t i o n  -17-  may  be made. Then by  ^J  i s spanned  q  and  by  j  the  above argument  elements  q  and  (2.10),  X J X  of t h e f o r m  . . x 1,  '2 1  hence ,q , .q n — n J  for  alln  (1.2), j  Using Witt's formula as f o l l o w s .  - rf  T =  where t ^ , t ^ , . . . q  i s the  be  given  Let  (3.2.1)  j  may  q  sum  q  n (i - t r  a r e commutative  of t h e  ,  v  i n d e t e r m i n a t e s . Then  coefficients  of a l l t e r m s  t ^ ' t . . . t ^ i n t h e f o r m a l e x p r e s s i o n o f T as a power x  k  series,  such  that  ^1  +  ^"2  " " "  +  11  2 2  In  particular,  (3-2.2)  i  q  of  = j  there  i =  elements  a.{a.  to  element  and  an  each  (1 < n < p-1). —  j  £_ ;  q  i s t h e number of  a^,  q  involve  a ,..., 2  a^ taken  Among t h e s e just  one  combinations p at a  combinations  g e n e r a t o r a^  taken  1, 2,..., q, t h e s e c o r r e s p o n d t o t h e  p times,  each  '  = 0  repetitions.  are q which  hand,  q  c a s e n = p,  allowing  P  n  t h e g e n e r a t o r s of  time  =  k k  n In  +  a.  ...  a . / l 1...  l ] = 0.  On  of t h e r e m a i n i n g c o m b i n a t i o n s a. f a . i, i  submodule  ^j(  x  a. i  ... 3  the  other  corresponds  a, / I 1 ... ip  1]  £ 0  and  ), n + n + . . . + n 1  2  =p,  -18-  of  I  i s s p a n n e d by one  J p  Therefore  they are l i n e a r l y  (3.2.3)  gives  one  i n d e p e n d e n t and we  may  easily  have  "  1  an e x p l i c i t e x p r e s s i o n We  of t h e s e e l e m e n t s .  (''T ' " * ' ^ T ^  q - i l - * -  0  This  and o n l y  of t h e rank of  e v a l u a t e t h e n e x t few  Jfp'  i ^ .. p +i  J  In  c a s e n = p+1,  t^'t^...t^  In  il.,  P+2  t^t^  o 1  -  and we  p  P  p  -  +  f  4  j  "  U  q  ( "  2  f  1  p  P-1  IO 3 J  2  q  ^ v r  2  ) ( lp-2 * - ) f  p  ^ 2  .  Thus  ^  i  6  C q  +  "  2  q  H  *  q  t  t  2 t  obtain  ^  h  ^  p  &  3 q  }  p-2  (  }  4)( p. 3 ) q  q +  - tj */ - 3 2 i " - 2  of the  p-2  of t h e c o e f f i c i e n t s 2  3  we  &  j  -p-(d -q)(q -q 2)(q 2  + 1  2  and  of t h e c o e f f i c i e n t s  p-1  M  and  condition,  - ^( ^ ; ) •  , i s t h e sum P+4  have  terms; t ^ t ^  -  - X  1  f  of t h e term  have  ID 2 3 , t^tgt^ and t^t^  j '  +  choice  f3 ?( l* - ) *  -  p+3.  j  j  a r e two  i ^ „ i s t h e sum  (3.2.3),  Next,  there  one  s a t i s f y the r e q u i r e d  (3.2.3), 2  terms  and we  1  c a s e n = p+2,  Similarly,  i s only  = t.fP" ,  K  2 o— 2 ^"2 i which  v r  there  3  a n d  P  4  of the t  4 t 2  terms  i " p  t  S  ^  p-  -19-  <3-*-3>4  =  "*P4 +  f  5^^p-l ^ 2  f  4 2^ f  3  q  q  3  q +  + ^ < q - q ) ( q - q ) (q -q*2) 3  +  38ir  2  2  ( q 2  -  q )  (  q  2  ~  q  +  2  An u p p e r bound i(n^ ng,...,n f  (3.2.4)  i  negative to  t ( V  k  p  3  2  2  V  2  over  q  )  5  for  by a s i m i l a r method. L e t  q  ) ) "  ) a r e commutative  i s taken  integers  q+ p  j(n^,n^,••.,n  T» - H ( l - t ( v , V  the product  ( _' )  (<* -<^) ( q - q + 6 ) ( ^ - ) _  )  ) may be o b t a i n e d  where t ( v ^ , V ^ , • • • , V and  ^^^^^^p-^^  +  ^-( -q)(q - 3)( ^2 ) 3  +  +  f  (  l '  V  V  2 V  indeterminates  a l l q-tuples  V^, V^,..., V^. D e f i n e  o f non-  j(n^,n ,...,n 2  be t h e sum o f t h e c o e f f i c i e n t s o f a l l t e r m s  ll' 12'---' lq  such  V  V  ) M  -  • * ^  \  l  '  \  2  °  that A,^ + ^2 V l l  +  ...  +  V 2 1  +  = p,  +  •••  +  V k l  =  n  l'  A.. v.. + A. v + ... + A. v. = n. , 1 lq 2 2q k kq k ' 0  then  we  have  0  ,  f  T  t  )  -20(3.2.5)  i(n It  (3.2.6)  1 #  n ,...,n^) < j ( n ^ , , . . . , n  i s noteworthy  that,  i f r > p-1,  elements  j(r+p-2,2,0,...,0)  which  are e i t h e r  i s t h e number o f  o f t h e form x ^ r x ^ a^ ...  1 ...1 j  \l or o f t h e f o r m x j a  a\ , 3| l a l ... **• l ) I 1 1 .. . 1 J Q  elements  respectively element  then  j(r+p-2,2,0,...,0) > f(r+p-2,2,0,...,0).  By t h e d e f i n i t i o n ,  basis  ).  2  of  a n  of  a  1  a  where x^ and x^ a r e  1  (m^ , 1, 0, . . . , 0)  and  + m^  and  ^(m,,, 1, 0, . . . , 0)  = r , and x^ i s a b a s i s  ^ ( r - l , 2, 0, . . . , 0) . The number  of elements  r +2 of r  the f i r s t into  form  is [ — ]  ( t h e number  not more t h a n two p a r t s ) ,  elements Hence  of t h e second form  is  of p a r t i t i o n s of  and t h e number o f  f(r-1,2,0,...,0) = [—].  j(r+p-2,2,0,...,0) = [ ^ ] + [ f ] = r + according  as r =  0 o r 1 (mod 2).  f(r+p-2,2,0 Thus we have p r o v e d Since  0)  1  or  r,  On t h e o t h e r hand,  = [ "P"J--] . r  (3.2.6).  i(rVj,  ) , we \have  i(r+p-2,2,0, . . . ,0) < j(r+p-.2,2,0, . ..,0) if  r > p-1.  (3.2.7)  This  relation  iJJ < jJJ  implies  (n > 2p - 1)  -21-  rr2 4.  The i d e a l  be  d e a l i n g with  of  aL  will  I . In t h i s s e c t i o n we s h a l l 2 2 o n l y ju , J , and C . The g e n e r a t o r s 2  be d e n o t e d  by a and b . Thus  d e n o t e t h e submodule o f X. combinations with  o f homogeneous m o n o m i a l s  4-1. every  ( ^ ) n  t h e module this  £^(m,n)  m  a  together  JLXn^n).  By ( l . l )  i n a and b, and t h e  consisting  o f m a and n b  introduce a linear  s e t of normal p r o d u c t s  (4.1-1)  be t h e r a n k s  i n a l l p o s s i b l e orders  X_(m,n). We  =  y be w r i t t e n as a l i n e a r  of normal products  normal p r o d u c t s  multiplied  of  ^(m,n)  respectively.  Normal b a s i s o f  element of  combination  for  i n a and i n b  , and f(m,n) and i(m,n) w i l l  c£„(m,n) and JJ(m,n)  m  of a l l l i n e a r  d e g r e e m i n a and n i n b. S i m i l a r l y 2  ,£_(m,n)0 J of  consisting  X_(m,n) w i l l  span order i n  as f o l l o w s :  xaD < ybD,  any x, y and D, where D i s e i t h e r  E o r any p r o d u c t  t h e i n n e r d e r i v a t i o n s A and B. f o r example, ba < ab, 2 2  babab < ba b . T h i s i s a l e x i c o g r a p h i c o r d e r i n g right  from  to l e f t . Let  (4.1.2)--  x, < x 1  v  be the be  n  < ... < x.  2  l  a l l the elements of t h e ( * ) normal p r o d u c t s , m  order  d e f i n e d as above,  the s m a l l e s t i n t e g e r such  element  i n (4-1.2),  n  i n ascending t h a t x.  with  order. Let i  i s a non-zero  and i ^ be t h e s m a l l e s t i n t e g e r  -22-  such that i^  <  x.  , x.  i2*) 1  every on  x.  x^,  i ^< '  integer  j <  i ^ +i <  x. . Then lk.  we  x.  x.  be  i^., is linearly  does  .  (Clearly  such  that  independent elements  define  depend that  smallest  linearly  on  i . ., i s d e f i n e d k+1  (4<l>l) i n the sequence  There w i l l  be  and  dependent  i , ., t o be t h e k+1  not  . It i s clear  by t h e o r d e r i n g step.  on x.  l e t i ^ , i^,...,  z  uniquely each  linearly  are l i n e a r l y  such t h a t  x, , x, i« i  not depend  general,  R  x.  x. i( '  in  does  f(m,n) o f x.  which  (4.1.2) form  IK  a basis basis the  o f t h e module  elements  x.  expression  combination the  normal  £.(m,n). We  the normal  form of  basis  normal Then  and  the elements  rest of -  subsequence  has  either  first  that  basis  of z^ <  z  2  linear  be  called  z^, t = f ( m , n - l ) ,  basis  z b r <  of  -C(m,n). of  The  elements  v  z^ <  ••• <  z^ i s a  z^.  basis  element  t h e f o r m y a o r t h e form y-^a, y a , . . . , y a  be  first  where r = f(m,n) ' '  T  ••• <  g  are the  *L(m,n) c o n s i s t s  of  A normal  ... < y ,  g  ... <  <  <  ... < y a  the sequence  2  as a  and  <C(m,n-l) r e s p e c t i v e l y .  n  of the normal  Proof.  these  £(m,n),  1  •«• <  l » ) and  z'b < z*b < 1 2  f ( m - l , n ) and  <  2  2  of the normal  the form  m -  z  y^a < y a <  f(m-l,n) elements  of  elements w i l l  Let y. < y  z^ <  JL(  b a s e s of  x of  call  x.  Lemma 4«1« s = f(m-l,n),  basis  f o r an e l e m e n t  of normal  shall  z b . We  comprise  of  ^(m,n)  will  a l l the  show normal  -23-  basis is  elements  a normal  o f |_(m,n) o f t h e form y a . F o r i f y ' a  basis  element  some i , 1 < i < s, s i n c e  of J L ( # ) m  t h e n y ' = y^ f o r  n  o t h e r w i s e y' = 2 J J  y ' a = 2. y.. a and y ^ a < y ' a w h i c h  y^ < y ' and t h e r e f o r e says t h a t  y,' a c a n n o t be a n o r m a l  Moreover,  y,a, y a , . . . , y a are l i n e a r l y  because  o t h e r w i s e we c o u l d 0  X  basis  element.  0  have  0  <c <c  X  S  independent,  a relation,  + ... + 7^ y a = 0  A,,y,a + A , y a  where  V  (X,. £ G F ( p ) ) ,  S  X  or. ^ l This  l  y  ^"2 2  +  implies  y  •"'  +  that ^ ^  y  l  +  element  of t h e form  y s  ^*2 2  z e r o o r a, b o t h o f w h i c h Similarly,  ^  +  y  s^  a  °'  =  "*"  +  +  contradict  we may  ^s s ^ ^^her y  S  the assumption.  show t h a t  zb i s o f t h e form  a normal  basis  z,b f o r some i , I  1 < i < t. We basis all  shall  elements  normal  denote  o f £_(m,n)  basis  t h e s e t of a l l t h e n o r m a l by U(m,n) and t h e s e t o f  elements of  £_(m,n) o f t h e form zb  by V(m,n). The number o f e l e m e n t s is  f(m,n) - f ( m - l , n ) and w i l l Obviously,  «C(0,1),  be d e n o t e d by  the normal c L ( 2 , l ) and  oC(l,l),  of t h e s e t V(m,n)  bases of JL(1,2)  ^(m,n).  JL(l,0), consist  of the  2 single  element  We may g i v e U(k,2)  a, b, ba, ba  explicit  and U ( k , 3 )  formula  (1.3),  f o r the elements  of U ( k , l ) ,  f o r a l l k.  Obviously By  forms  and bab r e s p e c t i v e l y .  U(k,l) consists  o f one element ba*".  -24-  V(k,2) one of  <p(k,2)  = f(k,2)  ^ ( k , 2)  = 1  - f(k-1,2)  =0  i f k i s even and V ( k , 2 )  i s vacuous  0 (mod  2),  k == 0 (mod  2) ,  k =  consists  ba b i f k i s odd. T h e r e f o r e U ( k , 2 )  element  the [ e l e m e n t s  o f t h e form b a ^ b a ^  j  i s even,  is linearly  prove that  dependent  consists  ^, j = 1,  3,--'  i  •j I n d e e d , we may  of  ba ba  , where  on s m a l l e r  elements.  For 0 •=  ba (ba )  =  ba [BA ]  n  n  n  n  = ba BA , n n = ba ba n  (^)ba ABA ' r \Tl-L_ n-1 (^Jba ba  n  n  n  + ... + ( - l ) b a A B , / ,\n, 2n, + ... + ( - 1 ; ba b.  1  n  n +  n  n  n  or, , 2n, /'rMu 2 n - l , ba b = [^jba ba -  / \ n - l , n, n ... + ( - 1 ; ba ba , j  and a l l t h e e l e m e n t s on t h e r i g h t (4.1-1)/  than the element Again,  shall  show t h a t  elements  by  on t h e  (1.3),  V(k,3)  we  are s m a l l e r ,  i n the  ordering  left.  have  consists  i n the ordered  n  <j>(k,3) = [~^-] .  of the f i r s t  We  <jp(k,3)  s e t of e l e m e n t s y.b, y, £  U(k,2).  k+2 Consider It  has t h e f o r m b a b a m  and m > 2n+3, Therefore is m >  n  3 =  --|)  >  0.  set U ( k , 2 )  of t h e o r d e r e d  f o r an element b a b a b , m  r > n+1,  ba (ba r  n = k - m  2  f o r m - 2n -  U(k,2).  t h a n t h e [-^y~]th e l e m e n t has t h e form  2n+3«  m = n+l+r,  of  where m = 2 [ — j - ] + 1,  e v e r y element  greater  + 1 s t element  the [——]  n + 1  and c o n s i d e r  )(ba ) n  -  n  the r  ba ba ,  m >^ 2n+3, l e t relation:  -ba ((ba )(ba n  which  n + 1  )).  m  n  -25-  We  may  rewrite n+ 1  i f we  again write  products is  (-l)  smaller  ][BA ]  i n the  ba [BA r  and  this  = -ba [BA ][BA  n  n  2 n +  ^ba ba b m  this  and  n  than b a b a b . m  m  Therefore  U(k,3)  ba ba ba  where m  r  It  consists  first  <P(k,h) e l e m e n t s  The  first  and t h e s i m p l e s t  the  construction  but  ba ba ba b  M(2,)  }  V(l0,4)  denote  In J  2  = M(/\),  side are  elements,  V(k,3). form  that  we  can-  V(k,h)  by  taking  q  5  8  For a given  confusion,  ring,  we  set  ( ( i lj j 2 2 x  X  x  x  X  (1  \]  p X  ...  -'  x  £  of  the  symbol spanned  s h a l l use of t h e  the  elements  [x^}. we  have shown  U(m,,n.) f o r  1  and n , ) . ¥e have u s e d theorem  2  V(l0,4).  and y e t b a b a b a b e  set f .  3  bab > ba ba ba b  t h e submodule of t h e r i n g  '  some m.  normal  that  smaller  of the  where baba  the p r e v i o u s s e c t i o n e  on  contained in  o f more u s u a l n o t a t i o n  A=  of  on t h e r i g h t  dependent  e(x^) f o r the set c o n s i s t i n g  instead  r  f r o m y,b, y e U(k,h-l). J 3 c o u n t e r example o c c u r s i n  of V ( l 0 , 4 )  2. . In o r d e r t o a v o i d  x^,  ] [ BA J .  of t h e l e f t  have shown  of a ( L i e o r a s s o c i a t i v e )  will  symbol  element  method of c o n s t r u c t i n g  4 . 2 . •-• The elements  n + 1  i n terms  an u n f o r t u n a t e f a c t  the  t  n  i s odd, m < 2n+3 and m+n+r = k.  i s rather  2  + b a [ BA  of e l e m e n t s  continue t h i s  3  ]  identity  i s not  not  5  + 1  a l l the elements  2n+3  n  n  T h e r e f o r e we  n  1. e., b a b a b , m >  n  last  2n+3 i s l i n e a r l y  n  m  r  i n a and b, t h e l a r g e s t  ba ba b, m > m  form  3-1,  which  that  by  -26-  asserts may X  always  1^ 2  is  an element  be w r i t t e n  ""* p ^  X  o f t h e form y ^ { y ^ ••• y /1 ... 1  as a sum o f e l e m e n t s  *"* ^ "  X  m  a  i  n  to e s t a b l i s h another set  using to  that  a method w h i c h  could  purpose  of t h e form  of t h i s  such t h a t  be d e s c r i b e d  section  M(f) = J  -2  by  as " o p p o s i t e ' *  t h e e a r l i e r one. Theorem 4-2.  L e t I = £/x,(X~  ... x i D \  ~  ...  ( V  i J ) p  where (i)  e a c h x^ i s e i t h e r t h e g e n e r a t o r a o r  b _of JL^ o r a n o r m a l b a s i s element o f Xj^ o f t h e f o r m xb, i . e . an e l e m e n t o f t h e s e t V(m.,n.) f o r ' some m i' i i and n., l (ii)  i f x^[x  2  ... x  p  / l ... l } =  b {a ... a/1 ... l } t h e n e i t h e r D = E o r a p r o d u c t i n A and B, w h i l e i f x.fx„ b{a  ... x / l ... l l £  ... a/1 ... l } t h e n e i t h e r D = E _or D = BD», where,  again,  D  T  = E or a product Then M ( f ) = J The  into set  '  = e(x^{x2  x_^ s a t i s f y  -  p r o o f o f t h e theorem  two p a r t s . P  2  i n A and B.  In p a r t  I , we s h a l l  *"" p ^ ^ X  condition  will  prove that the  spans  D ED,. i.e.,  J  I I , we s h a l l  M(f') o ( p . M  , where t h e  denote  o f E and p r o d u c t s i n A and B by  In p a r t  2  ( i ) and D = E o r D i s a p r o d u c t  i n A and B. F o r c o n v e n i e n c e , we s h a l l consisting  be d i v i d e d  prove that  the set . Thus  M(r")sM( f ) ,  -27-  Proof '  P  a  r  !•  t  L  where each x^, i s a n o r m a l  e  n?::: :f)  £  t  basis  element  of  a  n  d  D e^Q.. O b v i o u s l y M ( f * ) = Suppose t h a t  t h e s e t o f p e l e m e n t s x^, x^,  . .., x^ c o n s i s t s o f i ^ e l e m e n t s y ^ , i ^ e l e m e n t s y ^ , .. i^  elements y^, i  (4.2.1)  x  il  x  +  2  '•*  11  •••  +  X  p l  ^ l j  =  . . .  1  i ^ . = P, t h e n  +  y  J  l  y  V  y  !  1  2 -  """- k\ y  2  1  j '  " " k i  i  (A. e G F ( p ) , A, * 0 ) , (2.10)  by x  lj) x 2 x  2  11  and ( 2 . l ) .  ... ""' x 1i and ° * l <j X  ... 1  J  a n  y  x, [x„ ... x / 1 1 2 p  if ...  1  replacing  y  Y l  \i  Thus by ( 4 . 2 . l )  M(Z.)  We Y  1  shall l  2y * "... " k y1 *  y  l  say t h e e l e m e n t s y  2  ~ l  i  ... i  11 D and X  any element  J  ec  *i u  v a  lent.  of t h e form  i s a s e t o b t a i n e d by  therefore,  one t h e n  we may  x,fx ... x / I ... l ] D o f P ^ b y 1 2 p e l e m e n t y^r y^ y ... y^JD w i t h o u t i - l i ... i j 0  2  1  loss  e  o f 2 , by an e q u i v a l e n t  an element  an e q u i v a l e n t  r  i s a s e t of e l e m e n t s  = M ( Z j . In what f o l l o w s ,  replace  a  2  k  of g e n e r a l i t y . We s h a l l  defined  divide  p  P(i,j;m,n)  into subsets  as f o l l o w s :  p(i/j; / ) m  n  =  e  /  y  y  i  a  /  2 """ i [ l ... 1  y  y  a  a  Z  l ^  *"* j k ^1^ 1 ... l m n Z  (i > P(0,j;m,n) = E / z ^ b j Z g b 1  ... z.b.a b D  J  ...  1  m n  a  1),  (j >  1),  -28-  f  (0,0;m,n) = e/aj a I fm-1  f  (0,0;0,p)  Then we  have  {J.  ivhere  r  1  denotes  integers  shall  now  a  y  U j ^ n F . - ^ j ^ n ) .  prove  that,  ""*  a  (2.2)  (4.2.3)  y a  y  x  v  j_| 2 v  2  a  relation  M(r').  =  a typical Y°  a  element  '""  Z  z  y.a l 1  a  **"  y  i  a  z.b 1 1  Y°  (i > 1). n  ...  z.b j ... 1  "''  Z  P(i,j;m,n);  of  a  m  Then by  =  i  y  f o r a l l non-  n w i t h the  1"'=  M(P*)  f 2 I i  and  taken  Define  Consider l  z  a  b  m  n  a  -^ 1  m+1  1 b  nJ -  v  i( 2 ^| 2 0 v  l  w  n >  J  the u n i o n  i , j, m  (4-2.2)  y  (m,  I^ 1J ,• j , m , n F ( i /' j ; n i', n )',  =  i + j + m + n = p .  We  = e/bfa b IDA ( |m n - l ) )  = f (0,0;p,0).  . i,j,m,n  negative  blD n)  a  y  a  "*" ...  2" 1  i '  y^(z^b)fy a 2  [  1  y  i  a  1  z  l ' 1 t  """ y° ... 1 z  z.b J 1  I  0 ... y ^ a ...  1  z^b  ...  0  . .  zz.b ,b J 1  a  b  m+1  n  a  b  m+1  n  a  b  m +1  n  1) ,  -29-  (Y ( 1 2  y,b|y„a 1 ) 2 i  may  be w r i t t e n  (4.2.4)  y  ...  i  z,b a J 0 m+1  ' .  a z h  1  ... y . a z.b ... z.b I 1 j ... i i ... i  (2.1l),  Using  y i 1  a  the f i r s t  a  b n-1j.  m+1  element  on t h e r i g h t  as f o l l o w s :  l( 2 y  y.a  a  z,b  z.b 1  I  1 -aiy  m+1  . y , a z.b ... z.b l 1 j 1 1 ... 1  a 2  I  a  3  n ) b] Y,  a  J  m  e M ( P ( i - 1 , j;m+l,n)) .. Writing y^(z^b),..., of  (4.2.3)  (4.2.5)  the elements y ^ ( y a ) , . . . ,  y^(z^.b),  y^b w h i c h  i n normal form y  l^ 2 ^j 2 y  a  y^(y^a),  2  y  i t may  " * i ... 1 0  a  I 0  a p p e a r on t h e r i g h t  y  a  l  Z  b  be seen  that,  "*" j k 1 ... 1 Z  a  b  \ n i  m+1  £ M ( P ( i - 1 , J ; m + l , n ) ) + M(P(i-2,j+1;m+l,n), (4.2.6)  y  1 ^ i^ ^) 2 1 z  >  y  **" ...  a  i  y  a  z  1  £ M(P(i,j-l;m+l,n)) (4.2.7)  y  l ) 2 b  y  1 e M(T  a  1  ""* ...  y  i  a  1  Z  l  have  (4.2.8)  M ( T ( i , j;m,n))  .  1  M ( p ( i , j-l;m + l,n)) + M(p(i-l,j;m + l,n))  particular,  m+1  k] n )  T(i-1,J;m+1,n)), z.b J 1  a b  m+1  \  n-1 \  + M(T(i-1,j+1;m+l,n-1)).  + M( T ( i , j ; m + 1, n-1) ) + M ( P ( i - 1 , j +1; m + 1, n-1) )  + M(p(i-2,j+l;m+l,n)). In  a  z  + M( b  (i,j;m+l,n-l))  Hence, we  l ^ * ° * y° 0 ... 1  -30-  (4-2.9)  M(F(i,0;m,n)) S M(r(i,0;m +l,n-l))  + M( f (  + M(f(i-l,l;m+l,n-l))  0;m + l , n) )  + M(P(i-2,1;m+1,n)) 1),  (n > (4.2.10)  M(p  (i,j;m,0))  ^ M(f (i,j-l;m+l,0))  + M(T(i-1,j;m+l,0))  + M(f (i-2,j+l;m+l,0)) (4.2.11)  (j > l ) ,  M(f (i,0;m,0)) (i-l,0;m +l,0))  £M(T  + M( P ( i - 2 , l;m + l , 0) ) 2),  (i > and when  1,  i =  (4.2.12)  M(f (l,0;p-l,0)) M(f (0,l;p-l,0))  £  (4.2.12),  To e s t a b l i s h  xa ( a and  we n o t e t h a t a i a™ £  \ = yb |  I A ) "  + M(f (0,0;p-l,l)). i f x = yba , t h e n  P ( 0 , 1; p- 1,  0) ,  (P-M  Y  i f x = b a , then m  xa  j  = b|  Now  suppose  number o f t i m e s , we (4-2.13)  a_^a  i > 1.  m  £  T  Applying  +  21s ( T  a  finite  ;  (i,0;ia»,n»)) + 2 > (  M  (4.2.8)  obtain  M ( P ( i , j ; m, n ) )  £  ( 0 , 0 ; p-1, l ) .  P ( i , J ; m * , 0) ) T  ZiM(r(i» j';m«,n')). f  i- < L  V, Again in  applying  the f i r s t  applying  (4-2.9)  summation  (4.2.10)  p(i,0;m',n ))  f o r each term M( of ( 4 - 2 . 1 3 )  f o r e a c h M(p  ifn  1  ?  > 1 and  ( i , j ' ; m ' , 0 ) ) of the  -31-  s e c o n d summation, we o b t a i n , (4.2.14)  in a f i n i t e  of s t e p s ,  M(P(i,j;m,n)) ^  2L M ( r ( i , 0 ; m t , 0 ) )  + 2L M ( r ( i  t  J ;m',n')). t  /  i.'<l  *A'  by ( 4 . 2 . 1 1 ) we o b t a i n  Again  (4.2.15) These  M(f  show;  ( i , j;m,n))  Z  S  M( P ( i » , j ; m ' , n ' ) ) . T  0 '.»  that  M(T(i,  f  Proof.  r  H  j;m n)) S  ,^  1  M(r(0,j';m»,n»)).  (4.2.2).  Hence we have p r o v e d  P  number  Part  II.  We s h a l l  divide  the set  = £/ x^( x„ ... x l D x. = a, x. = b o r x. £ V(m.,n.),' ( l( 2 " * ' p l ' l l l ' :/ ' D £ £1 . V I1 ... 1 :  into  X  X  X  subsets a c c o r d i n g t o t h e degrees of elements i n  a and b.  Let P(m,n) = T'n  and  7(m,n),  define P (m,n)D  = £(xD| X £ P  (m,n)),  where D £ _Q . By t h i s  definition,  P (m-l,n)A U P(m,n-l)B S However  p(m,n) may c o n t a i n p(m-l,n)A  elements  P(m,n). not b e l o n g i n g  to  either  of  t h e f o r m x , [ x ... x / l ... l | E . We s h a l l 1 2 p  the x^{xg  1  0  o r l ~ " ( m , n - l ) B , namely x  s u b s e t o f P(m,n) o f e l e m e n t s ... x  p  / l ... l ] E by /\(m,n).  decomposition  elements denote  o f t h e form Now we have a  o f p(m,n) i n t o m u t u a l l y  disjoint  subsets, v i z : (4.2.16)  P(m,n) = p (m-l,n)A<J P ( m , n - l ) B U A ( m , n ) .  -32-  Simi l a r l y , (4.2.17)  ;  p(m-l,n)A = [7 ( m - 2 , n ) A U P ( m - l , n - l ) B A U A ( m - l , n ) A . 2  We (4-2.18)  s h a l l p r o v e , when n > 2,  that  M( A ( m - l , n ) A ) £= M( T ( m - 2 , n ) A VJ r ( ~ l / n - l ) B A U A ( m , n - l ) B ) 2  m  = M ( P (m-1, n )A V j p ( , - 1 ) B m  We of  n  have t h e f o l l o w i n g  three  A(m-l,n)A).  types of elements  A(m-l,n)A:  Type I .  x, i x 1 2 (1  where x^, x^, . . .,  x^ a r e o f d e g r e e g r e a t e r  in  a and b, and i >  Type  II.  '*  Type I I I .  xa Using  for  0  ... x, a i ... 1 h  b\ A \ k) ,  , .-  P  /_ nr 2  ^A = y b a . P  (2.2) we have t h e f o l l o w i n g  X  l P 2  "  X  ll 2 X  - x,  . x  IX  lj 2 x  l| 2 x  blA  h  k>  a  1  h+1  1  X .  1  X .  0 x  a  x.  1  x  i  . 1  U  one  1, h < p - 1 .  an element o f Type..T,  (4.2.19)  than  1  X .  1  1  a  k )  M 2 X  h+1  k >  a  b)X.  h +1  k»  a  b  |B  h + 1 k-l)  relation,  -33-  Moreover,  by ( 2 . 1 1 ) ,  (4-2.20)  X j j x^ ... x^  I1  ...  x ( |x„ x 3 2  [l and  i f i > 2,  we have,  1  a  b|  h+1  k i  ... x.  a  ...  1  h+1  i f i = 1,  (4-2.21)  xif M = b I a b IX 1h + 1 kj (h+1 k-1j a  By w r i t i n g nation  each o f X^, X^,...,  of products  element  j = 1,  2  X^ as a l i n e a r  combi-  of ( 4 - 2 . 1 9 )  every  i s contained i n  U p(m- 1, n-1) BA VJ P(m, n - 1) B) , f o r X  2,..., i . For b>l| aa lm-1  an e l e m e n t o f Type b 1 laa = b( h( a n-1) (m  yba  P  P  Thus we have Using  b l j n-2]  b 1 n-2 \ ,  I I I may be w r i t t e n  = y(ba ) - ya b P  bb ) )Jb n-2}  b M - bl a n-2ij \m  •2bla (m and an element o f Type  I I , we have,  b ) - b( a n-lj (m  )/bf a = b/ { (m  readily  x  i n A and B, we can see t h a t  i n the r i g h t  M(p(m-2,n)A  = -ba y —  ya b.  P  P  proved (4-2.18).  induction  on b o t h m and n, we may  show t h a t  M(P(m,n)) ^ for  b)X. 1 kJ  1  a l l m and n, w h i c h  implies  M( r ) that  M(p) e ( T ) . M  This  completes the proof  o f Theorem 4-2.  t A,  -34-  4«3-  The u p p e r bounds  k(m,n).  L e t k(m,n)  be t h e number o f e l e m e n t s of ]~~ (m,n). Then by Theorem 4 « 2 , ,  (4-3-1)  i ( m , n ) < k(m,n). Now, by ( 4 . 2 . 1 6 ) ,  (4.2.17)  and ( 4 . 2 . 1 8 ) ,  we have (4.3.2)  T(m,n) = P  UP w h i c h a t once (4.3.3) Let  implies  (m-2,n)A U r ( m - l , n - l ) B A 2  (n,n-l)B  U4(m,n)  (n>2),  that,  T(m,n) =  P(m-j,n-l)BA U^(m,n). j  h(m,n) d e n o t e t h e number o f e l e m e n t s o f  Since /\(m.,n)  J  are mutually  (4.3.4) Thus  P ( m - j , n - l ) BA , j = 0 , 1 , . . . ,  the sets  disjoint, m.  k(m,n) =  k(m,n) w i l l Since  write  (4.3.4)  (4.3.5)  m, and  we have  k ( m - j , n - l ) + h(m,n)  i f we know k ( m j n - l ) f o r a l l m  finding  A(m,n).  f  (n > 2 ) .  < m, t h e p r o b l e m o f  be r e d u c e d t o t h a t  of f i n d i n g  h(m,n).  k(m,n) = 0 when m + n < p , we may  as f . o l l o w s : r-i  k(r+p-n,n) = ^  k ( j + p - n + l , n - l ) + h(r+p-n,n) (n > 2) .  In o r d e r t o f i n d consider Let i ^ , i^ > parts,  h ( r + p - n , n ) , we s h a l l  subsets of /^(r+p-n,n) d e f i n e d  first  as f o l l o w s .  i^,..., iy, where i ^ + ... i j , = n and > • . . > i j , > 1 be a p a r t i t i o n o f n i n t o k < p +  and d e f i n e ^ r ^ l ' ^  i  k  )  +  -35-  = e / x.. | x~ ... x, a i \ x, £ Vim , ,n . ) f o r some m,\ | 1 2 k I j J J J 1 V £ Note  1 p-kJ  ...  11  /  A(r+p-n,n).  that  i^,  ig,..*,  i ^ . denote  the degrees  ...  , Xj_ i n b r e s p e c t i v e l y . T h e r e  the  degree  of ...  i s no r e s t r i c t i o n  o f e a c h o f t h e x^ i n a e x c e p t t h a t  d e g r e e s o f x^, x^, • • • ,  number o f d i f f e r e n t  For a given  subsets A  integer  and t h e r e f o r e  does  number 7r(n,p) i s well-known  +  n, t h e  ( i . , i , .. . , i , ) i s the r 1' 2 ' ' k 0  number 7r(n,p) o f p a r t i t i o n s o f n i n t o not more p parts,  on  t h e sum  x^. i n a i s r - ( i ^ + i ^  + i ) + k = r - n + k .  x^,  o f x^,  n o t depend  than  on r . The  and i s u s u a l l y g i v e n by  the f o l l o w i n g g e n e r a t i n g f u n c t i o n : (4.3-6) Z T(n, )t = TT(1 - t ) " . n  1  1  P  n-o  l~\  Denote ^ r ^ l '  i  2 ' " " '' k ^ i  t h e number o f e l e m e n t s b  y  h  of t h e s e t  r ^ l ' 2 ' ' * " 'k^ * i  Lemma 4 - 3 • 1 •  i  i  F o r i ^ > 2 and j < p - k,  h(i^ i2,««"/i^»l/l/•••,!) #  -  h^(i^,i2f.../ij,).  Proof .  The mapping o f A  o n t o A ^ ( i ^, i 2 / • • • / i^.) d e f i n e d  gives  1  the r e q u i r e d  j  one-to-one  Thus t h e p r o b l e m  (X  lj  p-k-j.  ( i ^i^,  . . . , i ^ , 1, . . . ,  by X  ...  r  2  X.  ...  1  k  p-k  correspondence.  of f i n d i n g  (,ig,•••,i^)  -36-  with  i  i , > 2. k — we  1  > We  i s reduced s hia] all  q used f o r i n J  obtain  now  of h ^ ( i ^ , i ^ , . . . , i ^ )  use a s i m i l a r method  IT (1 m, n  - t(m,n))~  t lie , t (m, n ) a r e commutative  taken  over  is  a l lpositive  integers t h e sum  greater  1.  expansion  i s clear  o f T",  shall  '  n )  ,  indeterminates , m i s and n i s t a k e n  over  , • • • / i j, )  of a l l terms  i^.) i n t h e f o r m a l  power  s u c h t h a t m^+m + ... m^  = r-n+k  +  2  e v a l u a t e k ( r + p - n , n ) , n = 2,  3/  4«  that, 7(r+p-l,l) = -  and  ( m  Then h^ ( i ^ ,  of t h e c o e f f i c i e n t s  We It  integers  than  t (m^. i ^) t (n^, i g ) • • • T. (m series  to  f o r h^_ ( i ^, i ^ , • • • , ij,) • L e t  where  all  3,  J  T» =  with  t o t h e one  and j ( n , , n „ , . . . , n ) i n s e c t i o n 1' 2 q  an e x p r e s s i o n  (4.3-7)  to that  M(r(r+p-l,l)) M(ba  r + P  " ), 1  therefore.  (4-3.8)  i(r+p-l,l)  = k(r+p-l,l)  =  1.  Similarly, J(p-n,n)  and  we  = M( T(p-n,n))  have  (4.3.9)  i ( p - n , n ) = k(p-n,n) = 1 By  (4.3.5)  k(r+p-2,2)  and  (4.3.8),  = X. k ( j + p - l , 1 )  we +  = r + h(r+p-2,2).  (l< n < have h(r+p-2,2)  -37-  On t h e o t h e r h a n d , A(r+p-2,2) and  = A (2)  = e(xa  r  V(r-1,2)),  |x £  p - 1  therefore, h(r+p-2,2)  Cp(r-1,2)  = h (2) = r  = 1 or 0, as r £ 0 o r 1  according (4.3.10)  k(r+p-2,2)  according  0 or 1  as r £ Putting  (4.3.11) We s h a l l  (mod 2 ) .  or  r,  r = p - 3 we  have  = f(2p-5,2)  = p - 2.  show t h a t k(r+p-2,2)  which t o g e t h e r  = i(r+p-2,2)  (4-3-11)  with  J(2p-5,2) The  (0 < r <  p-2),  that  = ot(2p-5,2). uses t h e e x p l i c i t  e l e m e n t s o f j£_(r+p-2,2)  4 « 2 . Suppose  section  implies  of ( 4 - 3 . 1 2 )  proof  form o f n o r m a l b a s i s  consists  = r+1  (mod 2) .  k(2p-5,2)  (4.3.12)  in  Hence,  that  obtained U(r+p-2,2)  r i s e v e n . Then  of the f o l l o w i n g elements:  baba  r + P  ~  < ba ba  3  3  r + P  ~  < ... <  5  r / 2 e l e m e n t s and t h e l a s t  The  first  the  sequence  a r e o f t h e form x a  and t h e r e f o r e  they  bl a (p-2  obtained  r  P  2  linearly  ~ b. 2  r / 2 elements i n  and b a y P  respectively  t h e element  b\a = ba " ba l ]  w h i c h does n o t depend J(r+p-2,2)  contains  r + p  in ^J(r+p-2,2).  are contained  J(r+p-2,2)  Moreover,  P  ba  r  + bj a b|a (p-3 l )  r + 1  on t h e r e l e m e n t s o f  previously.  Therefore  ^J(r+p-2,2)  -38-  contains of  at  least  r+1  linearly  j£_(r+p-2, 2) , i n o t h e r  ( 4 . 3 . 1 2 ) . By in  case  a similar  of r i s By  set  and  A  (4.3.10),  we  (4.3«12)  prove  have 2 = f  according  A(r+p-3,3) (3)  can  proves  odd.  = 0 or ~  The  (4-3-10)  and  method we  r-i Z,k(j+p-2,2) where  r+1,  >  (4-3-1)  wi t h  elements  words,  i(r+p-2,2) which t o g e t h e r  independent  + a as  0 or  r =  c o n s i s t s of  the  1  2).  (mod  A (2,l)  sets  r  and h(r+p-3,3)  = h (2,l)  +  = h (2)  h (3).  r  +  r  h (3) r  r  Now, h (2)  = 1,  r  according = e(xa  p - 1  as  |  = 0,  1  (mod  2)  and  A (3)  since  r  V(r-2,3)),  x e h (3)  where  0,  r s  0,  =  <f ( r - 2 , 3 )  1  2  -  , -  =|  +  , according  a , 2  as  r s  0,  1,  2  (mod 3 ) . A l t o g e t h e r , by ( 4 - 3 . 5 ) , we have (4.3.13) k ( r + p - 3 , 3 ) = - ^ - ( 3 r + 2r + a), 2  where a = 6 , 3,  4,  5  (mod  1,  2,  3,  4,  -1  according  as  r =  0,  1,  2,  6).  This Meier-Wunderli  expression [22]  (4-3.13)  i n h i s study  on  was the  obtained  by  s t r u c t u r e of  -39-  the  B u r n s i d e g r o u p s . As we s h a l l  his  result  implies  i s , k(r+p~3,3)  i(p+r-3,3) The  i s not o n l y  but a c t u a l l y  = f(l2,3)  -  i(3p-9,3)  = f(3p-9,3).  1,1(13,3)  Again,by  j  4,  +  - 3 , 3 )  P  = 0, 7,  where 3,  = i(r+p-3,3)  ^ (2,l,l),  we  = i  + f  3  A (4)  = h (2,l,l)  (h)  = e(xa ~ | P  h (4)  =  r  where 3  0 or 1  r =  = £/ X J x  I and  1 = —^-  2  11  2  1  1 o r - -g— , a  (mod 2 ) . a p-2  iI  Again,  2  r  r  r  and  therefore  + P r + a , 2  2  = JL  as  Z\ (2,2) r  x  2  e V(m ,2), 2  therefore, r  r  + h (2,2).  ) I  h (2,2)  + h (2,2)  3 = 0 o r -g- a c c o r d i n g  E V(m ,2), 1  2,  so t h a t  + h (4)  + h (4)  2 = -§-  1,  c o n s i s t s of  x e V(r-3,4))  f(r-3,4)  as r = 0 ,  r  r  7,  f l  T  r  r  A  +  + h (3,l)  r  i(l2,3)  and A ( 2 , 2 ) ,  r  5•  have  7 according  = h (2) + h (3) Now,  f o r r < 2p -  and f o r p >  The s e t Z ^ ( r + p - 4 , 4 )  r  h(r+p-4,4)  = f(l3,3),  (4-3-13)*  A (3,l),  r  an upper bound f o r  f o r p = 7#  that,  4/ 3» 4 ,  5 (mod 6 ) .  6)  ( r < 2p - 5 ) ,  equality holds  a u t h o r has v e r i f i e d  X k (  (Section  that  k(r+p-3,3) that  see l a t e r  , 0 , J-  -  J~  ,  0,  m+ 1  m= 2  r-2  -40-  according  as r = 0 , 1,  (4.3-14)  2 , 3 (mod 4 ) - A l t o g e t h e r , we o b t a i n  k(r+p-4,4)  (4r  + r  3  2  + a r + |3),  where a = 16 o r 4 a c c o r d i n g as r = 0 o r 1 (mod 2 ) ,  12, 1$, 16, 15, 12, 7  P = 2 4 , 15, 4 , 15, 2 4 , 7, according  as r s 0 , 1, 4«4.  Then k k  p+r  2,  11  The upper bound k •  2  f o r r = 0 , 1,  5 and s h a l l  o f r t h e u p p e r bounds k  p  +  2  t h e u p p e r bound j obtained p+r  been u n a b l e although  2  t o prove  that j p+r  evaluate  show t h a t f o r t h e s e  are equal  r  k(m,n).  earlier.  respectively  The a u t h o r has  = k i n general, p+r ^ '  i t seems t o be t r u e . First,  we  note  Lemma 4 « 4 . 1 «  ... h(p,r)  A  = 21  Let k  i s an u p p e r bound f o r i . We s h a l l r  r  values to  (mod 1 2 ) .  i  = h ( p - l , r + l ) = ... = h ( r , p )  . 2' 'P'r'o•o fk' i  1 #1 /  Consider  ( l< r < p-l).  ijj. ^ / i  w h e r e t he subset  2  i  +  i  2  +  + i ^ _ + j = r + n, o f t h e s e t ZXp-n,r+n) . We s h a l l t h a t j > n. E v e r y in (i  element  of the subset  has t o t a l  degree ;  a and b a t l e a s t +l) + ( 2 ^ i  1  for then  +1  +  i s a non-zero i t has t o t a l  show  / +  ^k"*" ^  element  1  +  j  +  ( ~ p  k - J  ')  has t h e d e g r e e  degree at l e a s t  therefore p+r+n-j < p * r  =  P r+n-j +  i > 1 inb  i + 1 i n a and b .  -41-  and  thus  contains  we have j > n. T h i s f a c t only those  subsets  where t h e number o f ones On  A  shows t h a t  (i i  r  , • • • , i j , , 1, 1, • . . , 1)  i s always not l e s s  t h e o t h e r hand, by Lemma 4-3 •!»  (4-4-1)  2  A ( p - n , r + n)  t h a n n.  we have  h (i ,i ,...,i ,l,!,...,!) r  1  2  k  - h ^ ( i ^ , i , ...,1^,1,1, . . . , l ) , i  2  where t h e r i g h t of  the subset  Applying °f  A(p  - n  this  hand  s i d e i s t h e number o f e l e m e n t s  A ^ ( i ^ , i , . . . , i ^ , 1 , 1 , . . . ,1) 2  argument  /r+n),  t o each s u b s e t  of  A(p,r).  A ^ ( i ^ , i , . .. , i^) 2  we o b t a i n h(p-n,r+n) = h ( p , r ) .  Thus t h e lemma has been Lemma 4.4.1/  proved. together with  k ( p - n , n ) = 1 and t h e i d e n t i t y  the fact  (4-3»5),  implies  Lemma 4•4•2. k ( p , r ) = k ( p - l , r + l ) = ... = k ( r , p ) Using k  . p+r  readily.  this  lemma we can e v a l u a t e  k(p+r-2,2),  ... , k ( l , p + r - l ) .  We know, 6 ( k ) = 1, 1,  1.  p  (4-4.2)' while  k  = p - 1,  p  k(p,l) = 1 implies,  (4.4.3)  certain  Let 6(k ) denote t h e sequence p +r  k(p+r-l,l),  (4.4.2)  (l < r <p-l).  6 ( k  p+l  )  =  1  ' ' 1  * "  1  '  -42-  (4-4-3)'  k  p  +  = P.  1  A g a i n k ( p + l , l ) = 1 and k ( p , 2 ) = 3 ( c f . ( 4 - 3 . 1 0 ) ) (4.4-4)  (  6  k  (4-4-4)'  p  +  k  2  )  =  1  (4.4.5)'  3  '  3  = 3P -  p + 2  '  3  1  '  1.  by ( 4 . 3 - 9 ) ,  Similarly, (4-4-5)  '  imply,  (4-3-10)  6(k ) = 1, 3 , 6 , 6 , p+3 k = 6 - 4-  and ( 4 - 3 - 1 3 ) ,  6, 3 ,  1,  P  +  by ( 4 - 3 - 9 ) ,  Again,  ( 4 • 3 • 1 0 ) , ; iH.3 -13) and  (4-3-14) (4-4-6)  (  6  k  p +  (4-4-6)'  4^  k  =  1,5,10,15,15,-.-,15,10,5,1,  = 1 5 P - 13-  p + 4  Now A ( p , 5 )  of A ( 5 ) ,  consists  $  A ( 3 , l , l ) , A ( 2 , 2 , l ) and A  A (3,2).  5  5  5  5  A  (4,1),  (2,1,1,1).  Therefore h(p,5)  = h (5) + h (4,D 5  5  h (3,2)  +  5  + h (3,l,l) + h (2,2,l) + h (2,1,1,1) 5  5  5  = h.(5) + h (4) + 5  + h (3) + h (2,2) 5  5  = f (1,5)  h (3,2) 5  + h (2) 5  + <f ( 2 , 4 ) + ^ ( 1 , 3 ) 9 7 ( 1 , 2 )  + <f ( 3 , 3 ) + <f ( 2 , 2 ) ^ ( 1 , 2 )  +(j>(4,2)  = 1 + 1 + 1 + 1 + 0 + 0 = 4, 4  k(p,5)  =^k(j+p-4,4) j =°  + h(p,5)  = 30.  Thus we have (4-4-7) (4-4-7)'  °(  k  p +  k  $) p + 5  =  1,5,14,23,30,30,...,30,23,14,5,1,  = 30p - 34-  -43-  5-  Lower bounds f o r t h e c l a s s /J ,  be  the L i e q u o t i e n t r i n g  to  t h e L i e r i n g o v e r GF(p)  of  i.e. £  Q  £ .  Let  q  £  q  i s isomorphic  q  g e n e r a t e d by e,,  e ,..., 0  e  I d  q  1 in  which  pair to  the  of e l e m e n t s  specify  The  t h e exponent  ring is nilpotent  problem,  related The  1  x, y of the  [12]  nilpotence  proved that  gave a b a s i s finitely  shall  £, (5)  generated Engel  the  Meier-Wunderli w i t h prime less  than  nilpotent  class  [22]  has  exponent  of c l a s s  exists  of an  problem.  less  of t h i s  a positive  proved  than  13  and  that  any  but  he d i d  the B u r n s i d e  group  of  CJ^CP) cannot  2 p - l . (See  section  a ring  Recently,  be n i l p o t e n t  that  Kostrikin  4 over  is nilpotent,  shown t h a t  exponent  4,  of c l a s s  of n i l p o t e n c e .  g  unsolved  r i n g of exponent to 6  £, (p)  generated  Engel r i n g with  is nilpotent  implies  Theorem 5•1« there  i s a well-known  p > 3 cannot  2 p - l , which  main p u r p o s e  =  a finitely  of t h e r i n g . Higman [ 9 ]  determine  write  F o r t h e c a s e of exponent  c h a r a c t e r i s t i c prime  not  The  we  If i t i s necessary  to the r e s t r i c t e d B u r n s i d e  or 2 i s t r i v i a l .  of  p-1,  ring.  q u e s t i o n as t o whether o r not  Engel  = 0 h o l d s f o r any  i d e n t i c a l r e l a t i o n xy  also  i s to prove  F o r any  positive  integer  m such t h a t  class be  Green the  integer  followi d  i f p is  2 a prime cannot  g r e a t e r than m, be n i l p o t e n t Let  (4.3.I),  then  of c l a s s  the Engel r i n g less  e(r,n) = f(r,n) -  e(r,n) >  than  i(r,n).  f(r,n) - k(r,n).  We  (p)  dp. Then  shall  by prove  [3]-)  -44-  Theorem >  5«1 by showing t h a t ,  e(dp-n,n) > 0 .  k ( d p - n , n ) so t h a t By  Witt's  such  a polynomial  there  n-2 i n r  F (r)  1 +  R  r , and t h e r e  ( ^ ( r ) of degree  c o e f f i c i e n t s such  exists  another  n-2 i n r with  at most  n  (5-2)  that  <f ( r , n ) < <J) ( r ) n  a l l p o s i t i v e integers r. Since i s clear  <£> ( r ) has n o n - n e g a t i v e  ^ Using  (  n  such  a polynomial  )  ^ ^  n  (  r  +  1  )"  F o r a given  H  ( ) r  n  o  degree  r  integer  prove n > 2  there  a t most n - 2 i n r  that  (5-4) for  r  (5>2) and ( 5 - 3 ) we s h a l l  Lemma 5«3« exists  coefficients,  that  (5.3)  h(r+p-n,n) < H ( r ) R  a l l positive integers Proof.  r and p r i m e s p.  L e t 7r(n)  be t h e number o f p a r t i t i o n s  n, so t h a t  (5.5)  Z vr(n)t n=o  Clearly the  establish  n > 2  F ( r ) o f d e g r e e a t most  1  non-negative  of  integer  f(r,n) > J L r ^"  polynomial  it  For a given  a l l p o s i t i v e integers  for  we may e a s i l y  that  (5.D for  (1.3)  formula  Lemma 5 • 2 . exists  f o r some n, f ( d p - n , n )  n  =  f l (1 V =• i  t )" . 1  7f(n) > 7r(n,p) f o r a l l p, w h i c h  number o f s u b s e t s  A  (i , i , . . . , i r  1  implies ), i i  that  +i ~ 2  -45-  + ... + i  p r o v e Lemma 5«3#  partition  i ^>  such  Therefore  i t i s enough t o show t h a t  i ^>  nomial H ( i ^ ,  ... >  of n t h e r e  • • • r ij,) °f d e g r e e  f o r each  exists  at most n - 2  h (i ,i ,...,i r  in r  1  2  f c  )  < H (i .i , r  1  ,i  2  f c  )  a l l r. If  k = 1, i ^ = n,  A (n)  = e(xa " | P  r  therefore  h (n) =  ^ r ^ l '  \ However,  1  V(r-n+l,n)),  n  k > 2 and i  If  1  x e  5> ( r - n + 1,n) < $ ( r - n + l ) .  r  = £/x f x  1  then  (5-2)  by  (5.7)  1  ?  1  ... x  2  1  k  ... 1  noting  k  > 2,  then  )  a | j x. £ V(m  ,n ) , 1 < m < r - n + 1 \  p-k) I  that  /  t h e maximum of each m  d e g r e e of x^ i n a) i s r-n+1,  (the  i t f o l l o w s that the  number of a l l p o s s i b l e c h o i c e s o o f  e a c h x, £ V(m,,n,)  3 in  a poly-  that  (5-6) for  greater  7T'(h) f o r any p r i m e p. ( c f . S e c t i o n 4 - 3 - )  than to  = n, k < p, o f A ( r + p - n , n ) i s n o t  A  (i  i , . . . , i  ) i s not g r e a t e r  2  3  3  than  <f(l,i ) + < f ( 2 i ) + ... + f ( r - n + 1 , i , ) f  < ( r - n + l)  j  (by ( 5 - 3 ) )  ( r - n + 1)  j i f i , > 2, then k — h (i ,i ,...,i ) 1  Therefore (5.8)  r  x  2  k  < ( r - n + l) (J>. ( r - n + 1) <J> k  ( r - n + l ) . . . (|)  (r-n + l ) .  -46-  Now, t h e d e g r e e  i n r of the p o l y n o m i a l  k + '(ij-2) + ( i Finally,  -  2  2  )  (5.9)  h (i ,i , r  <  1  + ... + ( i j . - )  = n - k.  2  i f i . > ... > i . > i . , = ... = i 1 — — j j+1  by ( 4 « 4 « l ) and above  then  on t h e r i g h t i s  argument,  , i )= h (i ,i ,  2  k  r  1  ,i  2  f c  )  ( r - n ' + l ) <t>. (r-n»+l)<J). ( r - n » +1) . . . l 2  . (r-n» +  j  X  j  X  1  where n* = i . '+ i„ + ... + i , and t h e d e g r e e 1 2 j the p o l y n o m i a l  is j + (i^-2)  on t h e r i g h t  +  i n r of (i ~2) 2  + ... + ( i . - 2 ) = n - k. J Thus we have o b t a i n e d t h e r e q u i r e d  polynomial  ( i ^ , i , . . . , i ) and Lemma 5»3 has p r o v e d . 2  Proof  of Theorem  5-1-  by i n d u c t i o n , t h a t f o r a g i v e n a polynomial n-1  such  (5.10) for  K ^ ( r ) of degree  We s h a l l  show  first,  integer n there  exists  i n r not g r e a t e r  that k(r+p-n,n) <  a l lpositive  n-1 (n-1)'  + K(r)  r  n  i n t e g e r s r and p r i m e s p.  We have a l r e a d y v e r i f i e d 2,  3 and 4- Suppose  integers  than  less  than  (5-10) n. Then  holds  (5.10)  f o r n = 1,  f o ra l l positive  by ( 4 « 3 . 5 ) and Lemma 5 - 3 /  «"-' k(r+p-n^n) = ^ . k(j+p-n + l,n-1)  + h(r+p-n,n)  -47-  n-1 = ~~ rr (n-l)! Clearly, degree the  K  ^( ) r  +  Z  K n  _  1  + K (r) + n T  ( j )  i n r not g r e a t e r  +  (  H n  r  )  i  J  s  p o l y n o m i a l of  a  t h a n n-2, w h i c h  by  (5-1) and  1  r = (d-l)p,  + F (r)  n _ 1  n  n-1  , , - K (r),  ~ J^iji r  n  then  e(dp-n,n) >  _L-(dp-n)  -  -  n _ 1  +  F ((d-l)p) n  ((d-l)p) (d-l) ,  V N  n  _  1 r (  -  n _ 1  djii-l  - ((^Tt)  For integer  a given  n such  integer  1  d-1  f i n d m such t h a t e(dp-n,n) >  ,*  - n)p  •  E (p), n  in p less  d >^ 2, we may  -  n  than  always  n-1  find  > 0,  n satisfying this f o r any prime  n  completes  condition,  we  may  p > m  i ± i T ~ ' ((-±-) ~ nl  This  n  that  (-J-fh a v i n g chosen  K ((d-l)p) x n-1  where E (p) i s a p o l y n o m i a l o f d e g r e e  and  completes  (5.10)  e(r+p-n,n) > - ^ ( r + p - n )  an  K  induction. Now,  Let  ,(j) + H( r ) . n-1 n  K  J  1  - n ) / "  1 +  d-1 t h e p r o o f of Theorem  E(p) > n 5.1.  0.  -48-  6. denote or  B u r n s i d e s groups.  the free  "J (n)  usual,  the subgroup  g , g £  group ^  generators,  or simply  will  by l 6 ( n ) ,  disprove  /  exponent  ( n )  q  <  ^' (n) i s n o r m a l CJ  will  be c a l l e d ,  as  n with g w h i c h we  [2] i t to prove or  conjecture:  Any f i n i t e l y  satisfies  (n)  *26(n) o r 3 .  g e n e r a t e d group of  n, i n t h e s e n s e t h a t  group  q  g e n e r a t e d by a l l  q  o f exponent  problem  the following B^:  , -or s i m p l y *=f ,  t h e B u r n s i d e group,  q  Burnside's  the  q  t h e Burnside group  denote  of ^  • The s u b g r o u p  R  and t h e f a c t o r  q  g r o u p w i t h q g e n e r a t o r s and l e t  denote  elements  Let  T  every element  the r e l a t i o n  g  11  g of  = 1, i s f i n i t e .  T h i s may be r e p h r a s e d a s : B^ : The restricted  of  weaker p r o p o s i t i o n  Burnside conjecture R  an upper  (n) i s a f i n i t e  n  :  n that  be r e p h r a s e d i n terms  (cf. Let  Baer [ l ] , H a l l  "Y^ . be t h e i t h t e r m  a R  XMP)  a  group P  n  d  "O  = 0 °°  satisfying  is a finite  exists  of a l l f i n i t e  groups  p, t h e p r o p o s i t i o n  of  and Higman  i- = l  R  t h e B u r n s i d e group [7],  Higman  p  T£)(p) .  [9])  of t h e l o w e r c e n t r a l  *o  of  known as t h e n i s as f o l l o w s :  can be g e n e r a t e d by q element s.  When n i s a p r i m e can  R  F o r e a c h g and e a c h n , t h e r e  bound f o r t h e o r d e r s  exponent  group.  series .  ^  O  .. When p i s a p r i m e , 1  the hypothesis i n the p r o p o s i t i o n  p-group  and, as i t i s w e l l - k n o w n ,  -49-  nilpotent.  T h e r e f o r e such a group  13 (p)/I6>  f o r some r , and  rephrase R  as  P l  V> ( P ) / © o o  •If +  •••  = lS(p)/1S any  JMpj/J^pQ  1  q  of  o  i  s  2  q  B^ has Waerden [ 1 4 ] .  may  group.  that  i s ,i f  some r , t h e n  r  Z&Cp)/^^  i s known t o be  i  "|5(p) /TS^  trivially  2 is abelian  fo (2)  . Thus we  of  implies  finite  for  R. p  concerning this  following:  is  group  f  f a r , t h e known r e s u l t s  are the  exponent  is nilpotent,  7&co >  i , the n i l p o t e n c e  problem  is a finite  • Since d3(p)/C>  So  tf5(p)/K>  group  follows:  R :  r  so of  is a factor  true.  F o r , a group  and t h e r e f o r e  of  the o r d e r of the  . been  p r o v e d by  Levi  o r d e r of $ 3 ( 3 )  The  q  and  Van  i s shown t o  der be  j B  4  R^ q  has  been  p r o v e d by Sanov  has  been  p r o v e d by  = 2 and by Higman  B,- i s s t i l l  f o r any  [12]  Kostrikin finite  for  q. However,  undecided.  There and G.  [9]  [23].  are a l s o  Higman [ 9 ]  important r e s u l t s  on t h e f o l l o w i n g  still  of P.  Hall  weaker  propos i t ion : S an u p p e r of  n  :  F o r e a c h q and  each n t h e r e  exists  bound f o r t h e o r d e r s of a l l s o l v a b l e  exponent  n that  can be  g e n e r a t e d by q  groups  elements.  -50-  It the on  i s not  intention  above p r o p o s i t i o n s , Engel  rings  to the  6.1. and  the  facts  +1  begin  a n C  but  Burnside  rings.  the  We  between £^  representation  of  the  free  Let  OC.^  (  simply (X)  the  ring  A^,  over and  let  of  (or  same r i n g g e n e r a t e d ^  q  ( o r £>) w i l l  integers)  defined Again,  i = 1,  successively as  ^  Y. and  by  be  sl^I  a^,  of  2,...,  q,  [27],  The s$  i  and  xy — >  symbols & , n q  )  respectively,  denote  the  free  associative  has  k^,...,  A^,  a^. As  (over the  Lie  = XY  - YX,  X e 3  [XY], f)  q  • ky  ring  of  their  r3  , Y £  X,  £5  of homogeneous e l e m e n t s  shown t h a t  the  3.  the  (or simply q  of  products,  ^ n CX ,  1,  the  where x — *  submodules of  the  in section  and  =  series.  L i e r i n g over  • •• ,  of and ^^n q  power  free by  £, .  [18]  generators.  Magnus [ 1 6 ] power s e r i e s :•  the  consisting  d e g r e e n i n the  be  ring  [17],  l i n e a r combinations  i s well-known A  groups  formal  Lie ring  [XY]  a.. —>  factor Engel  [l6],  groups  well-known  generated  the  by  m a p p i n g 6: y —*  &J)  results  review  the  g r o u p by  denote the  consisting  e l e m e n t s A_,,  Magnus*  integers  our  Burnside  the  of  recall  o r  the  briefly  we  ring  with,  disprove  groups.  shall  submodules  or  to apply  between  connection  *  to prove  rather  Connection  Engel  about  ^n^^n To  our  formal  (X  , n  and of  $  q  -51-  g. = 1 + g  = 1 - &  1 i  the f r e e 6.1,  represented the  we  -  2  ... ,  group with  i n t h i s way.  ( i = 1,..., g) ,  q generators.  use °$ ^ o r  where D The  £^9  n  w  n  +D  n  , D n +1  to denote the f r e e  Each element  G £ °^  group  has  + D  n  1,  =  w  subgroup  2,..., f o r some n. '  consisting  of a l l e l e m e n t s  . + ... , D £ Jy , i s t h e n t h t e r m n+1 ' n n  If n  u  the lower c e n t r a l  the  ,+..., n+1 ' £ ( T .• i n +1  nth dimensional  1 + D  series  of ^ " ( [ 1 7 ] ) .  Moreover,  mapping:  nh". T  G = l + D GG»  + D  n = (1 + D n = 1+(D  , + . . . —> D , n+1 n + D . +...)(1 + D» + D' n+1 n n+1 + D») + . . .  n gives  —»  n  D  + D » , n  n  o7  where t h e k e r n e l  is  «T ,, i . e . ' n+1  n How, (of,  +...)  a homomorphism o f t h e m u l t i p l i c a t i v e g r o u p °^  on t o t h e module  [17],  (6.1.2)  consider  [18],  t , , / 6  abelian  the f a c t o r  [26]),  M  l  s  group  n  n  +  i  ' T . u ! ? „ n ? ( p ) )  n  Tn fn.l'rn.l(T nT(p))/J„. -  l3 /U^> n  group ^ 3 / o  Since,  T  = the  In t h i s  form G = l + D  of  + A  i  generate section  A.,  /  n  n+  ^ i  sa  homomorphic  1  image  -52-  of oL • L e t r be t h e k e r n e l , i . e . ^ n T n (6.1.3) (We  « » ' B ,  shall  use t h e symbol  a quotient  t  l  n  ~  * t „ - | , . instead  n  of  /  lt  to denote  n  module.) I t i s e a s y t o see t h a t  contains u  all  t h e e l e m e n t s o f p £.  element  then  of ^ G  there  = £(px| x £  n  f o r i f an  has t h e f o r m : 1 + pD + D , + ^ n n+1  exists  an e l e m e n t  1  +  D  n  +  D  n 1  G  £ T such ' n  that  + . . . ) ( 1 - D* + n  ...)  T  +  +  and (G'  _ 1  )  = (1 + pD  P  n  = 1' + D« 'n + 1  * " '  +  £  7  :  n+ 1  or  Thus, but  <  ^b ^ ^  also  n  1  n  +  ^  s  ^~  on  no  -'-y  image o f  of the f r e e  X,  Lie ring  Therefore,  (6.1.4)  13  t„  some submodule ^ ^ o f The  of  n  D  J  n  ?„•  •  1  X(Magnus  J"(Sanov [ 2 6 ] ,  the Engel i d e a l f  n  -  sum ^? = y  direct  known t o be an i d e a l  (6.1.5)  homomorphic  a  of t h e submodule  over GF(p).  for  i  * T 2  +  "  '  i  s  [ 1 8 ] ) and c o n t a i n s  Higman  [9]),  n = l , 2,  . . . .  -53-  Con s e q u e n t l y ,  (6.1.6)  , Or t  BJTB n  n+1  Whether o r not  - f n  = J  s  I  ^n  - J  = £  s n  n  . n  i s not known, e x c e p t f o r Sanov's  result :  (6.1.7) which  y  = J  n  implies  (6.1.8)  B  n  / B  Sanov's X  e  X 1  (n < 2p - 2 ) ,  n  n + 1  s [  proof  (n<2p-2).  n  of  (6.1.5) u s e s t h e f a c t  that  X  , e ,...,  e t generate a free  z  group  (Magnus [ 1 8 ] ,  [19]), together with the Baker-Hausdorff formula. Higman's p r o o f commutator  collection  process.  of (6.1.5) u s i n g  another proof tation  [6]  i s b a s e d on H a l l ' s  well-known 6.3,  In s e c t i o n Zassenhaus'  i n the f r e e Now,  (6.1.6),  by  i.e. f  nilpotent,  associative  =  £  ring  is the  n  =B  n  i f the E n g e l r i n g  £. i s  , = ... = 0 f o r some n,  ^ n n +1 = ... = fi^ and c o n s e q u e n t l y  + 1  also nilpotent Engel r i n g  question open.(cf.  and f i n i t e .  £  implies  o f whether  result  Thus  ' ^/  the n i l p o t e n c e  the p r o p o s i t i o n  o r not R' i m p l i e s P  R.  The  p  B' i s P  of  still  [l]). A few r e s u l t s  ^n^^n +l  of  over GF(p).  0  TQ  give  represen-  of t h e f r e e g r o u p s by f o r m a l power s e r i e s  elements  then  we  <  ?  e  n  e  r  a  l  a r e known about  exponent  the  subgroups  p. By c o m b i n i n g  (6.1.8) and o u r p r e v i o u s ones, we  have  Sanov's  -54(6.1.9)  rank(fo / B  n+1  n  calculus,  - j  n  J  ( l < n < 2p-2).  q  n  obtained  following results:  k  / B  p  p t l  the  p  +  )  1  -  -  /B , ) p  Fox  —  u s i n g the  ran (a  free  ("T^  differential  +  - (<) t ^  2  —  2  ) ,  rank(i6 . /iB .3)  = f  p  t  2  " 3p  • 1  ( P > 3-  q = 2),  rank(B  - f  p  +  3  - 6p  + k  (p >  q - 2),  p <  2  p  +  p <  3  /S  p  which c o i n c i d e w i t h The c o i n c i d e with  t  4  )  our  results our  lower  bounds.  of M e i e r - W u n d e r l i  lower  5,  bounds f o r t h e  [20]  also  ranks  of  (r+p-3,3) f o r r < p - 2. Using  7  —  q  Lyndon [ 1 5 ] ,  r « k ( B  6  J > f  f o r the  7 w i t h two possible  (6.1.9)  order  of t h e  generators.  result  by  we  any  may  o b t a i n a lower  Burnside  This  g r o u p of  i s , however, not  means.  bound  exponent the  best  -55* 6.2. order we  The f r e e r e s t r i c t e d  to g i v e ,  introduce,  s e c t i o n , the notion  restricted  Lie ring,  properties  of Zassenhaus'  free  The consists  dimensional  isomorphism  of £  onto  (6.1.5),  recall  some  subgroups  of a  [29].)  L i e r i n g ^$ , as d e f i n e d  o f e l e m e n t s 6x,  In  of t h e " f r e e "  and i n s e c t i o n 6.3,  ( c f . [10], [ 1 1 ] , [ 2 8 ] ,  group,  .  6.4, o u r new p r o o f o f  in section in this  ~j  Lie ring  in section  1.2,  x E «t , where 6 i s t h e  T9 d e f i n e d by,  6a. = A. , 1  1  6 ( y + z) = 6 y + 6 z = Y + Z , 6 ( y z ) = [(6y)(6z)] = [ Y Z ] = YZ - ZY. Thus, even  o9 does though  not c o n t a i n  these  elements  elements  enjoy  like  the  X , P  X e £  ,  property  P [YXX^TTx] = [ Y X ] = Y X P  -  P  a n a l o g o u s t o [ Y Z ] = YZ - ZY w i t h and a l l X , X £ now d e f i n e a new  X Y, P  Z = X . P  We  shall  P  Lie ring  $  which  does  contain  ^  [p l S  Let elements X 3 not  +  /d^ ^ P  P  +  direct  ^5  be t h e module  , X £ <\), and />3 <J^  P  sum.)  spanned  be t h e module sum  ^  • (Ofcourse,  In t h i s  module  /0  we  Lie multiplication,  this  namely  Zassenhaus,  t h e TT-oper at i o n , as f o l l o w s :  [ X Y ] = XY - YX,  sum i s  define  operations,  (6.2.1)  by a l l  two  and f o l l o w i n g  -56-  (6.2.2) By  X"= X . 77  (2.5) we have  (6.2.3)  immediately  [ X Y ^ ] = [Y""X] = [ X Y ] . P  Vie p r o v e operations.  that  X, Y £ $ *, i t s u f f i c e s  when  [XY] £ $  t o show t h a t  and X , Y  P  [X'Y»] = - [ Y ^ ^ Y ^  Thus we have plication. T  17  for  [X»Y»] e ^ *  £ -0 . But, u s i n g  (2.5)  show t h a t  (6.2.4)  X  t h e s e two  f  X' = x j , Y» = Y  we may  under  [ X Y ] , X " £ ^)  i n order to prove  e if  i s closed  It i s obvious that  X, Y £ ^) , so t h a t , any  P  £ 3*  seen  that  Next,  -  ]  1  i s closed  we u s e i n d u c t i o n  for X £  -2  x = x  ^  +  + x  n  £  C  p  +  Lie multi-  on e t o p r o v e  that  Let  +  + ... + x n  p S  0  ...  ]  under  p e  1  (  where  X s  J  0  +  /  j t p ]  X^, « • • , X^ £ Then by  X" = X 17  By i n d u c t i o n  x  ' ? x  0  e  of  i  i  ^ [ p " ]  m  will  + X  P  n  S  )  £  3  Lie ring be c a l l e d  characteristic By  p S+1 +  ...  +  X  p S+1 +  /\(X ,X  h y p o t h e s i s X^ = X^ £  x  The defined  i  (2.9)  (6.2.5)  A (  e-1-  +  '  t  h  u  s  ^  x ? r  £  p 6  )  and by (6.2.4)  "5 *•  w i t h t h e 7T-operat i o n t h u s  the f r e e  p (with  X  p 6  Q  q  restricted  Lie ring  generators).  t h e above argument  i t may be seen  that  -57-  is  t h e module  s p a n n e d by  a l l e l e m e n t s U\  , where  U, a r e b a s i s e l e m e n t s of *J and e = 0, 1. 2,... . l We s h a l l shox-r i n s e c t i o n 6.5, t h a t t h e s e e l e m e n t s U, l o* a c t u a l l y form a b a s i s of nj . Assume f o r t h e moment t h a t t h e s e t £(U. ) l the  form of  a basis  ^)  Then  f o r ^5  spanned we  have  not  (t  a l l elements of the set £ ( U  r$  i s closed  under  under t h e i r - o p e r a t i o n , e i t h e r of t h e s e Let  that the  |e>l).  P  J *=  that  under  .e submodulf  denote the  a d i r e c t d e c o m p o s i t i o n of t h e module  (6.2.6) Note  by  . Let  ^  L i e m u l t i p l i c a t i o n but  and &  i s not  operations.  denote the Engel  i s , t h e image of t h e E n g e l i s o m o r p h i s m 6:  closed  ideal  of J  ideal  o5 ,  of  L> u n d e r  £, — * ^5, o r , a l t e r n a t i v e l y , t h e TO"™ 1  module s p a n n e d by By  (2.8)  by  a l l elements  We  now  Engel  may  define ideal  an  a l l e l e m e n t s [XY  considered (X + Y )  . Let  of a l l l i n e a r  X ,  . Using  X £ ^)  Theorem 3«1 section ^  6.3,  - X  ^  consisting P  £ /-J , and  (cf.  [ll]).  i  P  /\(X,Y),  , w h i c h we may be  X,  Y £ ^ .  call  t h e submodule  of  the ^  t h e same method as i n t h e p r o o f  be  i n the e a r l i e r  shown t h a t  , i n the sense t h a t ,  [XYJ  - Y =  P  spanned  c o m b i n a t i o n s of e l e m e n t s  and t h e method i t may  t o be t h e module  of 3  ideal  of  P  ] , X, Y £ ^) .  X £ ^  i s closed  Moreover,  part  i s an  and Y £ J under  of  ideal  of this of  implies  the Ir-operat i o n  i t i s c l e a r that  ^  ^  and  -58-  we  have t h e  direct  (6.2.7)  %* =  (6.2.6)  By  (6.2.8) Let  £,  ^7  - ^  £  ) and  be  and  (6.2.7),  y  - ^  the  q u o t i e n t module  £  = J  ring  (6.2.8)  be  have  ring,  ^  ^  -  %  f o r %, <5  under t h e  (in fact, i s an  i s the  ideal  of  image of  i s o m o r p h i s m 6: £  —V  w r i t t e n as,  (6.2.9) in  we  , so t h a t  of  may  module  J - -J .  - ^  £  of t h e  ^ + fc\  is a quotient  the Engel Then  decomposition  £ , T  other  words, t h e  two  Engel  rings  £  and  Q  , the Q *  one  o b t a i n e d from  are  and  the  other  obtained  from  ,  isomorphic.  In t h e f o l l o w i n g , we s h a l l use symbols l i k e S , \ or t t o d e n o t e t h e submodule of t h e c o r r e s -v n ? n J n ponding all  module w i t h o u t  homogeneous e l e m e n t s  as we  have done  Zassenhaus g, i  =  (where A^  We  q  over shall  [29]  1  dimensional  has  shown t h a t  the  -1  g, i  =  1  use  - A.+ i  generators  GF(p),) generate  of  generators,  or  the  2  of t h e free  s u b g r o u p " of  subgroups.  f o r m a l power  A.l free  ^ , consisting  . . , q) , '^ '  associative  group with  denote  series:  1,2, . ''  (i =  to denote t h i s  ? n* "dimensional  i n the  Zassenhaus  the  again  of d e g r e e n  which c o n s i s t s  The  + A,, i  are  subscript,  earlier.  6.3-  Q(  the  generators.  group i n  the of  q  ring  nth  a l l elements  -59-  of  the  form G = l + D  subgroups  D  n  . + ...  (D.eOL.) i i on t h e s e d i m e n s i o n a l  n n+1 r e s u l t s of Z a s s e n h a u s  The main  then  + D  are the  following:  6.3.1.  If G =  £  n  1 + D  +D  + . . . £ „  n+1  n  , n'  n+1  0  ^  n  n  More p r e c i s e l y , t h e m a p p i n g : :  1 + D  G =  V  —f  + ...  D n  n GG»=  (1 + D  + ...)(l  + D  n  = 1 + (D gives ^  + D») n  n  T*  ^ n  p the  powers o f e l e m e n t s  )  following  (6.3.5) (6.3.6}.  series  or  and  the group of  T-^»  e /fc . n  n  l B  i s f " , "n + 1  "the i t h t e r m  a l l the of the  of Hf „ T  =  f f l  IT^P )]  (ip  3  we  J  have  > n, the  relations:  f*ilj"  >  the k e r n e l  g e n e r a t e d by  of t h e s e p r o p e r t i e s ,  n  + D* , n  J  Because  B  n  ( i P > n),  J * n  and t h e f o l l o w i n g  (6.3.7)  o n t o J) ^ n  6.3.4.  isomorphism  D  —>  ...  J  {^"^(p" )j denotes  lower c e n t r a l  +  = i^i(P )}  where J  ...)  n  a homomorphism o f  6.3.3.  +  T  n  n?(p).  ? "/*  n?(p».  homomorphism r e l a t i o n :  <~$:/j; c?>7(p))^ /s tl  n  n+1  .  i >  m).  -60-  (6.3-8) where  TP, /f) n  n +1  t h e submodule  ^ J  of Q * n  f  "  n  n  i s the kernel  of t h i s  n  homomorphism.  6 .4 • We  Connection  show, i n t h i s  which w i l l  give  section  another  To p r o v e if  D  6.4,  and  ^.  that  proof  (6.1.5).  of  (6.4-1) i t i s enough t o show  i s a monomial  n  between ^  of  and i f G = 1 + D  a n  ip  = n  j  where it  that  following,  zation  associative  H,  -1  2  elements  G £ f  (fT 0 n  ( w h i c h may  1  2'  r i n g over  generated  If  T(p))•  Now  be r e g a r d e d  X , X_,.... X n  of J .  p  J  D  ^ , n  n  = UP  as a g e n e r a l i -  (2.3)). be e l e m e n t s  GF(p) and l e t 0|  by f o r m a l power  series  W  Q  = W (H _,H 0  ]  H )  2  p  =  of a  free  be t h e  free  H  = 1 + X^,  (H H ...H ) ; P  1  2  p  then W  Q  where  each  least  p+1  = 1 + (X±  + x  2  + ... + X )  of t h e r e m a i n i n g  p  terms  i n t h e X,. L e t l W = 1 + w + w» , Q  1  P  J  consider  2 -,X. + X, - ... , i = 1, 2,..., p. L e t  =1  £ It ^ n  i n 6.2, , P /- * U £ fe, ^,  ... U / l 1 ... l } £  of J a c o b s o n 's i d e n t i t y Let  group  U  1  t h e U, a r e b a s i s k  i s clear  the  1, o r {U  j >  f  +...  n  T(2 n ' j ( p ) ) . As we have seen n +l n j> * e v e r y monomial o f i s e i t h e r of t h e form  that  + ... ,  i s of d e g r e e  at  then  -61-  where w^ i s t h e power each  of which  does  series  consisting  not c o n t a i n  X  and w" c o n s i s t i n g o f  n  1 the  r e m a i n i n g terms,  Then  i t i s clear  of a l l terms,  1  i . e . e v e r y term  o f w^ c o n t a i n s X^  that  1 + v  = ¥ ( l , H , . . . , H ),  1  0  2  and  W (1,H Q  Hp)" = 1 - w 1  2  2  + w  2  Let W  = W (H ,H ,  1  1  1  X  p  Q  1  1 is  p  + w*(-w^ + w^ - ...  P  where w which  i s t h e power  2  H p  )  _  for  contains P  let  ,  series X^,  do n o t c o n t a i n  terms. C l e a r l y , Once  + w^  2  ,  2  - (X„ + X_ + ... + X ) 2 3 P  t h e term o f l o w e s t d e g r e e . A g a i n ,  = 1 + w  0  ).  e v e r y term o f t h e power s e r i e s  and (X, + X_ + ... + X ) 1 2 P  n  2  2  = 1 + w* Note t h a t  H ) = ¥ ( H , H , . .-.,H )W (l,H  2  consisting  o f a l l t h e terms  and w^ c o n s i s t s  e v e r y term  of the remaining  o f w^ c o n t a i n s  b o t h X^ and X .  again, 1 + w  = W (H ,1,H^,...,H ).  2  1  1  p  Now l e t W  2  =  W (H ,H ,...,H ) = 2  1  2  = 1 + w^ + ( ~ 2 w  w  +  w  2  Every  term  W (H ,H ,...,H )W (H ,l,...,H )"  p  of the  1  1  2  p  1  2  contains  X^ and X , and t h e t e r m  of lowest degree i s  (x  (x + x  -  1  + x  (X  2  + ... + x  + X  p  )  p  -  + . . . + X ) p  p  2 ~ """ ^"  power s e r i e s f o r ¥  2  1  2  P  + (X  3  3  + ... + x  + X 4*  p  )  both  p  + ... + x ) p  j  1  1  -62-  Repeating t h i s W.  - W  +1  ^ i ^1'  =  H  p  i + 1  (  H l  i s , defining  ,H ,....H ) 2  p  2 ' * * "' i ' " " " ' p ^ i H  inductively, ¥  process, that  H  we f i n a l l y  p  + ZL(x +-x + . . . + x  0  x„ + . . . + x  0  c 1 (2-3)/  or u s i n g (6.4-2)  2  W  = 1 + fX  )  ^  1  x + ... +  2  0  1  ... X \ + V,  1  . . . 1 >  least  p+1 i n t h e X^ and moreover ... , X  series  V i s o f degree at  e v e r y term  contains  among i t s f a c t o r s .  p  now, i n t h e p r e v i o u s f r e e  G = 1 +  X .)' p-1  P  X  i n t h e power  Suppose,  Z(x,+  p  e v e r y term  2  -  p  ) - ... +2Lx +  where  o f X^, X ,  H  p-2  VI  all  H  see that  = W (H, ,H„, . . . ,H ) p. 1' 2' P = 1 + (X.+ 1 2 1  ' 2 ' * * "' ' * " '' p ^  W  1 2 1 1  + ... e  p ... 1  where e a c h U, i s a b a s i s  element  k  of  f  group ff, n  and i  i  k  n  1  +i  2  + ... + i = n. Then, l e t P G  = 1 + U  k  and c o n s i d e r W (G-^, G '; . . . , G ) . By !  p  2  W (G ,G , p  x  p  G ) = 1 + |U  2  p  U  1  I1  G  1 '  G  2  G  p  }  . . . 1J  1  i s of degree  ?n T<P)'  £  n  therefore, GW  (  P  G  i' 2 G  G  } P  1  £  . . ., p),  (6.4 = 2) we have  n i n t h e g e n e r a t o r s o f 0{ , o r  V  2,  ... U l + ...  2  where e a c h o f t h e r e m a i n i n g terms than  (k = 1,  + ... e '*f *  fc  T +1 n  ,  greater  -63-  or  (6.4.3)  G e J  7*n  and we have c o m p l e t e d The  l(p))  t h e p r o o f o f (6.4-1)•  above argument  the f o l l o w i n g well-known *ft[H H  H  . . . H  enables  formula  ]  over  ],H,  2  1  3  1  (mod T ( p ) T  X  p  +  J ,  commutator i s taken  i„, i „ , . . . ,  i o f 2, 3»•• • , P# P a r e g e n e r a t o r s o f t h e f r e e group % . In *c  and  t h e group  ] and t h e p r o d u c t  ] , . . . ] , H .  X  a l l permutations  (Grun [ 4 ] ) :  =1  where [ H I H J H . . . . H , ] d e n o t e s 1 1 2 X3 P [[.•-[[H,,H.  us t o g e n e r a l i z e  the  _p  part i cular [ H H H . . .~H ] s 1  2  Let  2  (mod T ( P ) T  1  2  G^, G ,..., G 2  be e l e m e n t s  P  G. = 1 + D. + ... e ^ k  Then  k  i t may e a s i l y  TTCG.G  3  I  P  =  where n = n^ + n  V  G  1 '  2  |  +  ...  2  ll  D |]  ... 1  JJ  lD 2 •"• p ] ••• 1 1 1 ... 1 J  D  D  ...  +  +  ^ n '  E  + ... n . On t h e o t h e r hand  2 V  G  1  "  1  +  )  l2  D  ••• ... 1 I  D  1 1  D  1  pl  +  ll  £  Therefore, F  G  l  G  i  9  2 i 3• • • i p^ p  t  G  G  G  (  G  l '  G  2  G  )  _  1  £  P  Tn 1 ' +  or TT  "  l l G  G  2  i • • • i G  3  ]  that  that  ...G. ] « 1 + [ DJ D  6• 2  such  ) •  (k = 1, 2,..., p) .  k  be s e e n  of %  P + 1  3  P  1  (  m  °  d  Tn  +  l ?  (  p  )  )  -64-  6.5set  e(U,  1,  already e  (U_^  are  2,..., f o r m  linearly  U^  use  induction  Suppose we  are b a s i s  l  only  on  1,..., e-1,  may  n  r  are elements  assume, w i t h o u t  o t h e r words, we  1 #  U.. i  elements  independent.  may  ... + Z ^  +  p  £ GF(p).  of a f r e e  associative  l o s s of g e n e r a l i t y ,  ring  that  i n e a c h g e n e r a t o r of  G[ ,  OlU^  ,n  ,...,n  ) fora l l  (2^ .u.  + ...  + Z>  P  P 2  P g  n  ^  l i i  (£\  .u  and  n  p  ^  + ...  ,u. ~~) p e  ei i  + 2>  l i i  ^  .u-  ).  p S  ei i  (6.2.4), W £ o9 .  V = y^-, l -U. i l + ... 1  + ^  e i'  u  pe-  1  then V z  = 0,  £  ^ (2.9)  p S i  i  Let  -  u  fixed  ±  w=  e i  f o r some  2  Again, l e t  elements  assume t h a t  n ,..., n , V  Then by  of  had  U i s a homogeneous e x p r e s s i o n in  the  are l i n e a r l y  A-,.  we  and  independent.  e, and assume t h a t  li  A  these  the  have  by t h e e l e m e n t s  are l i n e a r l y  U  U\  . S i n c e we  to prove that  the U  k Since  that  of /J  elements  for $  u = Z \ ) i i +Z*. u.  (6.5.D  p r o v e now  independent.  , f = 0,  p  We  i s spanned  Certainly We  .  a basis  d  seen t h a t  ), i t remains  P  for  ), wh e r e t h e U  P  1  e = 0,  A basis  0C(nf  _ 1  ,  n f "  1  n q  p  6  _  1  ),  p  q  -65-  and U = 2 A,„ . U. Oi  We have 0  - W + V . P  1  therefore = [UV] = [ ( Z A .  0 i  U.  - W)V] £ ^ ,  or  0 = (2^  0 i  u  - w)V £ Xs.  i  Since W and V have d i f f e r e n t generators,2L ^ O i ^ i ~ ^ ^ ^ ^ hence -  a n <  degrees i n the - W = 0  or  V = 0 . I f ~>A, .U. - W - 0 t h e n U = V ^ Oi I  By  induction  n  li'  '  Thus U ,  hypothesis, V = 0 implies  ei  we have c o m p l e t e d  of Let  f n  q  are l i n e a r l y  p  f  of the f r e e  n  q  *= Z  n Similarly,  a l l the . Oi  i n d e p e n d e n t and  e n a b l e s us t o d e t e r m i n e t h e r a n k s  be t h e rank o f ^  (6.5-2)  that  induction.  fact  submodules AJ  = 0 and V =  z e r o and a l s o W = 0 . T h e r e f o r e  are  ,.. ., U^  This  p  p  R  l e t f*(n^,  n  f ,  restricted  Lie ring.  (with q g e n e r a t o r s ) .  .  q  k  n/p^  1iv  n,j,...,n ) be t h e rank o f t h e  submodule ^) ( n ^ n j , . . .,n ) o f $ homogeneous e l e m e n t s  of degree  i n A . Then q q (6.5.3) f*(n ,  c o n s i s t i n g of  n^ i n A^,  i n A^, .  n  n ,..., n )  1  = 21  f  Then  2  ( i/p / n  k  n„/p ,..., k  n/p ). k  -66-  7. questions, arise  Unsettled  naturally  in connection We  j ( n ^ , , •• ',n^)  have  with  seen t h a t  the upper  1 2  by Lyndon  0  bounds  f o r i ( n ^ , , • • . ,n ) a r e a c t u a l l y  j(n n, + n  t o answer,  t h e above.  i ( n ,n ,...,n ) f o r t h e s p e c i a l c a s e s  calculated  for  following  w h i c h t h e a u t h o r has been u n a b l e  7.1-  to  problems. The  l f  previously  and M e i e r - W u n d e r l i .  n ,...,n  + ... + n  Similarly,  Is i t t r u e  ) = i(n^,n^,...,n  2  equal  that  ),  2p - 2?  < q —  i s i t true  that  k(m,n) = i ( m , n ) , for m + n <  2p - 2?  7.2.  We  have  verified  that  j(m,n) = k(m,n), f o r m + n < p + 4-  I t may  also  be e a s i l y  seen  that  j(r+p-n,n) = k(r+p-n,n), for  n = 1, 2, 3/  4-  Is i t , i n g e n e r a l ,  j(m,n) = 7-3. is  i t true  As  true  that  k(m,n)?  a generalization  o f Sanov's  theorem,  that  7-4«  The  a n a l o g u e of Theorem  5.1  f o r the  Burnside groups: For  a given  always p o s s i b l e of  nilpotence  positive integer  to f i n d  d,  a prime such that  of the Burnside  group  ig i t the c l a s s  jJ'/H'Cp) w i t h  -67-  two  generators  is greater  Certainly true,  but  not  7-h  were t r u e ,  necessarily  conversely.  I t seems t o  forms  However, we  dp?  i f 7.3  7«5explicit  than  f o r the  may  be  difficult  normal b a s i s  r a i s e the  w o u l d be  to  determine  elements  following  also  of  .  the  ring  question:  — 2 £_j  Let of  integers  product  if  generated 1  ba "^b^  1  Lie  a and  b.  r i n g over For  m  by  . . .a' b'* WK  , the  Kk  b,  in  normal basis  Lie  l e t x denote  the  . Then for  normal  free  riK  a and  a given  X-~,  is i t true then  ring  that,  e(u_^) i s a  for This  for  free  by  n  generated  e(u_^) i s t h e  basis  the  x = ba" 'b ' . ..a *'b ' of  o v e r GF(p) element  be  submodules  proposition  is true,  *C(m,n), when n =  1,  as  we  2 and  have 3-  seen  -68-  BIBLIOGRAPHY  1. groups  R.  Baer,  of a group,  The h i g h e r  Bull.  commutator  sub-  Amer. Math. S o c . v o l . 50  (1944) PP. 143-160. 2. in  the theory  W.  Burnside,  On  an u n s e t t l e d  of d i s c o n t i n u o u s  groups,  question  Q u a r t . J . 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