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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]

Let G and A be finite groups with A acting on G by automorphisms. In this paper we introduce the concept of "good action"; namely we say the action of A on G is good, if H = H , B C H (B) for every subgroup B of A and every B -invariant subgroup H of G. This definition allows us to prove a new noncoprime Hall-Higman type theorem. If A is a nilpotent group acting on the finite solvable group G with C G (A) = 1 , a long standing conjecture states that h (G) ⩽ ℓ (A) where h (G) is the Fitting height of G and ℓ (A) is the number of primes dividing the order of A counted with multiplicities. As an application of our result we prove the main theorem of this paper which states that the above conjecture is true if A and G have odd order, the action of A on G is good and some other fairly general conditions are satisfied. [ABSTRACT FROM AUTHOR]

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]

In this paper, we classify all finite groups satisfying the following property: their conjugacy class lengths are set-wise relatively prime for any six distinct classes. [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: 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]

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]

Abstract: Let A be a finite abelian group that acts fixed point freely on a finite (solvable) group G. Assume that is odd and A is of squarefree exponent coprime to 6. We show that the Fitting length of G is bounded by the length of the longest chain of subgroups of A. [Copyright &y& Elsevier]

Erratum to . [ABSTRACT FROM AUTHOR]

In this study we have obtained some sufficient conditions for the Taketa inequality namely dl(G) ≤ |cd(G)| for finite solvable groups G. [ABSTRACT FROM AUTHOR]