Abstract
Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M∖F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C F (H) = 1 or |C F (H)| =p and C F/C F (H)(H) = 1. Suppose that M acts on a finite group G in such a manner that C G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C G (H))+ 1. Moreover, the Fitting series of C G (H) coincides with the intersection of C G (H) with the Fitting series of G.
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Notes
Communicated by A. Turull.