On Engel rings of exponent p-1 over GF (p)

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.Science, Faculty ofMathematics, Department ofGraduat