 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 On Engel rings of exponent p1 over GF (p)
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
On Engel rings of exponent p1 over GF (p) Chang, Bomshik
Abstract
It is well known that the restricted Burnside problem for a prime exponent p can be rephrased in terms of the nilpotence of finitely generated Engel rings over GF(p),with exponent p1. We study these rings with the object of extending our knowledge of the Burnside groups. Let E(q) be the Lie ring over GF(p) generated by e₁, ..., e(q), where the elements of E(q) are restricted by the Engel condition [ fgP⁻¹] = 0 for all f, g ℇ E(q). If L(q) is the free Lie ring over GF(p) generated by a₁, ..., a(q), and if I(q) is the ideal of L(q) generated by [xyP⁻¹] for all x, y ℇ l(q), then E(q)≃L(q)/I(q). We study E(q) by investigation of I(q) in L(q). Let Iq/n be the submodule of 1(q) consisting of linear combinations of monomials in a₁,…,a(q) of degree n, and let I(q)(n₁.... ,n(q)) be the submodule of Iq/n consisting of linear combinations of degree n₁ in a₁, n₂ in a₂,..., n(q) in a(q), n₁ + ... + n(q) = n. The ranks of I(q) and I(q) (n₁ , ... ,n(q)) are denoted respectively by iq/n and i(q)n (n₁,... , n(q). We provefirst that I(q) is themodule spanned by all elements of the form [xyP⁻¹], and obtain upper bounds for iq/n and j(q)(n₁,...,n(q)) which may be most conveniently expressed as coefficients of certain formal power series. Further results are obtained by giving another set of elements which spans I². This enables us to find upper bounds for i²(m, n) by an inductive method. In particular, we prove [formula omitted] where K is a polynomial in r of degree at most n2. Using the aboveformula, we prove that, if the Engel ring E² were nilpotent with class c(p), then c(p)/p would not be bounded. Finally, we give a new proof of the relation between the Burnside groups and the Engel rings by studying the free restricted Lie rings and Zassenhaus’ representation of the free groups.
Item Metadata
Title 
On Engel rings of exponent p1 over GF (p)

Creator  
Publisher 
University of British Columbia

Date Issued 
1959

Description 
It is well known that the restricted Burnside problem for a prime exponent p can be rephrased in terms of the nilpotence of finitely generated Engel rings over GF(p),with exponent p1. We study these rings with the object of extending our knowledge of the Burnside groups.
Let E(q) be the Lie ring over GF(p) generated by e₁, ..., e(q), where the elements of E(q) are restricted by the Engel condition [ fgP⁻¹] = 0 for all f, g ℇ E(q). If L(q) is the free Lie ring over GF(p) generated by a₁, ..., a(q), and if I(q) is the ideal of L(q) generated by [xyP⁻¹] for all x, y ℇ l(q), then E(q)≃L(q)/I(q). We study E(q) by investigation of I(q) in L(q).
Let Iq/n be the submodule of 1(q) consisting of linear combinations of monomials in a₁,…,a(q) of degree n, and let I(q)(n₁.... ,n(q)) be the submodule of Iq/n consisting of linear combinations of degree n₁ in a₁, n₂ in a₂,..., n(q) in a(q), n₁ + ... + n(q) = n. The ranks of I(q) and I(q) (n₁ , ... ,n(q)) are denoted respectively by iq/n and i(q)n (n₁,... , n(q).
We provefirst that I(q) is themodule spanned by all elements of the form [xyP⁻¹], and obtain upper bounds for iq/n and j(q)(n₁,...,n(q)) which may be most conveniently expressed as coefficients of certain formal power series.
Further results are obtained by giving another set of elements which spans I². This enables us to find upper bounds for i²(m, n) by an inductive method. In particular, we prove [formula omitted] where K is a polynomial in r of degree at most n2.
Using the aboveformula, we prove that, if the Engel ring E² were nilpotent with class c(p), then c(p)/p would not be bounded.
Finally, we give a new proof of the relation between the Burnside groups and the Engel rings by studying the free restricted Lie rings and Zassenhaus’ representation of the free groups.

Genre  
Type  
Language 
eng

Date Available 
20120216

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080656

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.