On p-groups of maximal class as groups of automorphisms

Abstract

Let P be a p-group of maximal class and M be a maximal subgroup of P . Let α be an element in P \ M such that |C P (α)| = p 2 , and assume that |α| = p. Suppose that P acts on a finite group G in such a manner that C G (M ) = 1. We show that if C G (α) is nilpotent, then the Fitting height of G is at most two and C G (α) is contained in the Fitting subgroup of G. For p = 2, without assuming that C G (α) is nilpotent, we prove that the Fitting height h(G) of G is at most h(C G (α)) + 1 and the Fitting series of C G (α) coincides with the intersection of C G (α) with the Fitting series of G. It is also proved that if C G (x) is of exponent dividing e for all elements x ∈ P \ M , then the exponent of G is bounded solely in terms of e and |P |. These results are in parallel with known results on action of Frobenius and dihedral groups.