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 pc ≠ rd + 1 for any c = 1, 2 and any prime r where r2d+1 divides |G| and if CG(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 A1, a subgroup of A, where A1 centralizes D(R), then all irreducible characters of A1R which are nontrivial on Z(R) are computed.All nonlinear characters of a class two p group are computed.
