NILPOTENCY, SOLVABILITY, AND THE TWISTING FUNCTION OF FINITE GROUPS

Let G be a finite group. For every natural number m, we define the twisting function tau : G(m) -> G(m) by tau(x(1),..., x(m)) = (x(1)(x2), x(2)(x3),..., x(m)(x1)). It is clear that for each normal subset C of G, the function tau : C-m -> C-m is well defined, for each natural number m. Let C be a conjugacy class of size n in G. We prove (Theorem 1) that C is contained in the Fitting subgroup if and only if tau : C-m -> C-m is a permutation, for all 1 <= m <= 2n. We prove further (Theorem 3) that if tau : G(2) -> G(2) is a permutation then G is solvable.