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 non-identical 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/[Lj, Li, Lk<./em>] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[Lj, Li, Lk, Ll] is torsion free. We then prove that the analogous groups, namely F/[γj(F),γi(F),γk(F)] and F/[γj(F),γi(F),γk(F),γl(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 2-torsion in the Lie rings L/[Lj, Li, Lk] 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) = mT(m,k). We prove that if G is a finite m-generator group of exponent 8 then |G| ≤ T(m, 7471), improving upon the best previously known bound of T(m, 888).