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On Engel rings of exponent p-1 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 p-1. 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 prove-first that I(q) is the-module 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 n-2. Using the above-formula, 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.

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