Title:

Problems in Lie rings and groups

We construct a Lie relator which is not an identical Lie relator. This is the first known example of a nonidentical Lie relator. Next we consider the existence of torsion in outer commutator groups. Let L be a free Lie ring. Suppose that 1 < i ≤ j ≤ 2i and i ≤ k ≤ i + j + 1. We prove that L/[L^{j}, L^{i}, L^{k}<./em>] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[L^{j}, L^{i}, L^{k}, L^{l}] is torsion free. We then prove that the analogous groups, namely F/[γ_{j}(F),γ_{i}(F),γ_{k}(F)] and F/[γ_{j}(F),γ_{i}(F),γ_{k}(F),γ_{l}(F)] (under the same conditions for i, j, k and i, j, k, l respectively), are residually nilpotent and torsion free. We prove the existence of 2torsion in the Lie rings L/[L^{j}, L^{i}, L^{k}] when 1 ≤ k < i,j ≤ 5, and thus show that our methods do not work in these cases. Finally, we consider the order of finite groups of exponent 8. For m ≥ 2, we define the function T(m,n) by T(m,1) = m and T(m,k + 1) = m^{T(m,k)}. We prove that if G is a finite mgenerator group of exponent 8 then G ≤ T(m, 7^{471}), improving upon the best previously known bound of T(m, 8^{88}).
