Class two $p$ groups as fixed point free automorphism groups

Abstract

Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p c ≠ r d + 1 for any c = 1, 2 and any prime r where r 2d+1 divides |G| and if C G (A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|. The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A 1 , a subgroup of A, where A 1 centralizes D(R), then all irreducible characters of A 1 R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.