Action of a Frobenius-like group with fixed-point free kernel

Abstract

Abstract. We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that OEF; h D F for all nonidentity elements h 2 H . We prove that any irreducible nontrivial FH -module for a Frobeniuslike group FH of odd order over an algebraically-closed field has an H -regular direct summand if either F is fixed-point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F . Some consequences of this result are also derived.

Fitting Height of a Finite Group with a Metabelian Group of Automorphisms

“…Previously, it was proved in [9] that if C G B = 1, then many properties of G are "close" to the corresponding properties of C G A , possibly also depending on A. It is natural to try to extend these results to more general situations, for example, the condition C G B = 1 was relaxed in [3,10] under certain additional conditions, and the condition of BA being Frobenius was replaced in [4] by BA/ B B being Frobenius with B B of prime order, but keeping the condition C G B = 1.…”

“…The action of H on Q is coprime by hypothesis, and hence 2 Q = 2 Q H C 2 Q H Therefore, (4) and (5) can be combined to yield…”

On fixed-point-free automorphisms

“…After the present paper was submitted, the authors were informed by the referee about a recent paper by G. Ercan and İ. Güloğlu [4]. Here they consider a soluble finite group G admitting a 'Frobenius-like' group of automorphisms F R of odd order such that |F | is of prime order, C G (F ) = 1, and (|G|, |R|) = 1.…”

“…They are more general in the sense they do not require R to be of prime order, but less general in their stipulation that RF must be of odd order. Furthermore, the authors were also made aware that [4] contains Proposition C, which can be used to significantly shorten the proof of Theorem 1.2 of the present paper. This proposition is as follows.…”

Grupos finitos com um grupo metacíclico de automorfismos

“…Vale também mencionar que recentemente Ercan, Guloglu e Khukhro em [12,13] consideraram o caso onde um grupo nito G admite um grupo Frobenius-like como grupo de automorsmo. Um grupo AR é dito Frobenius-like com núcleo A e complemento R se A é um subgrupo normal nilpotente e AR/A é um grupo de Frobenius com núcleo A/A e complemento RA /A .…”