Fitting height of solvable groups admitting fixed-point-free automorphism groups

Abstract

C’onjectuye. Let G and A be finite solvable groups such that (’ G , d ,) 1, A f 1, and A acts fixed-point-freely on G. Then the Fitting height of G is less than or equal to the number of prime divisors (counting multiplicities) of .A . Thompson [I31 showed that even without the assumption of solvabilit!for G, if A is of prime order, G is nilpotent; i.e., the Fitting height of G is 1. Shult [I 1] showed that if A is a Frobenius group whose kernel and complement are of prime order, then the Fitting height of G is at most 2 provided that either 1 G I is odd or no Format prime divides 1 A I. Berger [2] has shown that if -4 is nilpotent and XX, wr Z, free for all primes p, the conjecture is also true. His result encompasses those of man!. others. The major purpose of this paper is to verify the conjecture, if certain conditions on the divisors of ~ G ! and 1 A 1 are satisfied, in the case that ;1 is a Frobenius group with cyclic kernel and complement of prime order. This result is contained in Theorem I .4 and Corollary 1.5 of Part II. The method of proof is basically that devised by Shult to handle the case in which the kernel of d is of prime order. In Part I, we pro\-e representation theorems for the semidirect product of a solvable group H by a proper subgroup A, of A of order prime to / H /. Shult’s results apply only if A, is cyclic; if not, we use Glauberman’s work on characters of groups admitting automorphism groups of relatively prime order. These representation theorems arc’ used in