Let P be a p-group of maximal class and M be a maximal subgroup of P. Let alpha be an element in P M such that vertical bar Cp(alpha)vertical bar = p(2), and assume that vertical bar alpha vertical bar = 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(alpha) is nilpotent, then the Fitting height of G is at most two and C-G(alpha) is contained in the Fitting subgroup of G. For p = 2, without assuming that C-G(alpha) is nilpotent, we prove that the Fitting height h(G) of G is at most h(C-G(alpha)) + 1 and the Fitting series of C-G(alpha) coincides with the intersection of C-G(alpha) with the Fitting series of G. It is also proved that if C-G(x) is of exponent dividing e for all elements x is an element of P M, then the exponent of G is bounded solely in terms of e and vertical bar P vertical bar. These results are in parallel with known results on action of Frobenius and dihedral groups. (C) 2015 Elsevier Inc. All rights reserved.
展开▼
