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Let A be a finite nilpotent group acting fixed point freely on the finite (solvable) group G by automorphisms. It is conjectured that the nilpotent length of G is bounded above by ℓ (A) , the number of primes dividing the order of A counted with multiplicities. In the present paper we consider the case A is cyclic and obtain that the nilpotent length of G is at most 2 ℓ (A) if | G | is odd. More generally we prove that the nilpotent length of G is at most 2 ℓ (A) + c (G ; A) when G is of odd order and A normalizes a Sylow system of G where c (G ; A) denotes the number of trivial A -modules appearing in an A -composition series of G. [ABSTRACT FROM AUTHOR]

Let 𝐴 be a finite group acting by automorphisms on the finite group 𝐺. We introduce the commuting graph Γ(G, A) of this action and study some questions related to the structure of 𝐺 under certain graph theoretical conditions on Γ(G, A). [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 subgroup H of a finite group G is called a TNI -subgroup if N G ( H ) ∩ H g = 1 for any g ∈ G \ N G ( H ) . Let A be a group acting on G by automorphisms where C G ( A ) is a TNI -subgroup of G . We prove that G is solvable if and only if C G ( A ) is solvable, and determine some bounds for the nilpotent length of G in terms of the nilpotent length of C G ( A ) under some additional assumptions. We also study the action of a Frobenius group FH of automorphisms on a group G if the set of fixed points C G ( F ) of the kernel F forms a TNI -subgroup, and obtain a bound for the nilpotent length of G in terms of the nilpotent lengths of C G ( F ) and C G ( H ) . [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: 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]

In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ CG(A) of prime order or of order 4, every conjugate of x in G is also contained in CG(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian. [ABSTRACT FROM AUTHOR]

Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p′-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible. [ABSTRACT FROM AUTHOR]

Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian. [ABSTRACT FROM AUTHOR]