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Let G be a finite solvable group and H be a subgroup of Aut (G). Suppose that there exists an H -invariant Carter subgroup F of G such that the semidirect product F H is a Frobenius group with kernel F and complement H. We prove that the terms of the Fitting series of C G (H) are obtained as the intersection of C G (H) with the corresponding terms of the Fitting series of G , and the Fitting height of G may exceed the Fitting height of C G (H) by at most one. As a corollary it is shown that for any set of primes π , the terms of the π -series of C G (H) are obtained as the intersection of C G (H) with the corresponding terms of the π -series of G , and the π -length of G may exceed the π -length of C G (H) by at most one. These theorems generalize the main results in [E. I. Khukhro, Fitting height of a finite group with a Frobenius group of automorphisms, J. Algebra366 (2012) 1–11] obtained by Khukhro. [ABSTRACT FROM AUTHOR]

Abstract Let FH be a Frobenius group with kernel F and complement H , acting coprimely on the finite solvable group G by automorphisms. We prove that if C G (H) is of Fitting length n then the index of the n -th Fitting subgroup F n (G) in G is bounded in terms of | C G (F) | and | F |. This generalizes a result of Khukhro and Makarenko [6] which handles the case n = 1. [ABSTRACT FROM AUTHOR]

A finite group is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup with a nontrivial complement such that for all nonidentity elements . Suppose that a finite group admits a Frobenius-like group of automorphisms of coprime order with In case where we prove that the groups and have the same nilpotent length under certain additional assumptions. [ABSTRACT FROM AUTHOR]

We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial F H-module for a Frobenius-like group F H 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. [ABSTRACT FROM AUTHOR]

Abstract: A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that is a Frobenius group with Frobenius kernel . Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide . It is proved that the derived length of the kernel F is bounded solely in terms of the dimension of the fixed-point subspace of H by . It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most , where r is the sectional rank of . As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to , a bound for the p-length of G is obtained in terms of the rank of a Hall -subgroup of . Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic). [Copyright &y& Elsevier]

Abstract: We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that for all nonidentity elements . We prove that any irreducible nontrivial FH-module for a Frobenius-like 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. [Copyright &y& Elsevier]

Erratum to . [ABSTRACT FROM AUTHOR]