Arnold D. Feldman

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

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

Persistent characterizations of injectors in finite solvable groups

Abstract In response to a question of Doerk and Hawkes [Finite soluble groups, de Gruyter, 1992, p. 553], we shall obtain characterizations of the injectors of a finite solvable group (without recourse to the concept of a Fitting set), and we also answer in the negative a question in [Dark and Feldman, J. Group Theory 9: 2006, p. 785].

Characterization of injectors in finite soluble groups

Abstract The purpose of this note is to describe when a subgroup of a finite soluble group is an injector of that group, without directly using Fitting sets.

Finite groups in which pronormality and -pronormality coincide

Abstract For a formation 𝔉, a subgroup U of a finite group G is said to be 𝔉-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉𝔉 such that Ux = Ug. If 𝔉 contains 𝔑, the formation of nilpotent groups, then every 𝔉-pronormal subgroup is pronormal and, in fact, 𝔑-pronormality is just classical pronormality. The main aim of this paper is to study classes of finite soluble groups in which pronormality and 𝔉-pronormality coincide.

Fischer subgroups, Fitting height, and pronormality

Several authors have studied the relationship betweeen Fischer

ℱ-Bases and subgroup embeddings in finite solvable groups

-subgroups and

Finite solvable groups in which semi-normality is a transitive relation

injectors of a finite soluble group. Fischer [6] proved that when

A class of generalised finite T-groups

is a special kind of Fitting set, now called a Fischer set of G, the Fischer

A nonabnormal subgroup contained only in self-normalizing subgroups in a finite group

-subgroups and the

Fitting height of solvable groups admitting an automorphism of prime order with abelian fixed-point subgroup

-injectors of G coincide. A bit more generally, Anderson [1] came to the same conclusion when the Fischer