Gülin Ercan

Finite groups admitting fixed-point free automorphisms of order pqr

Let G be a finite group and A be a group of operators of G with CGðAÞ 1⁄4 1. In [14], Turull proved that if ðjGj; jAjÞ 1⁄4 1 then (with certain exceptions for A), the Fitting height of G is bounded by the length of the longest chain of subgroups of A. We expect a similar bound for the Fitting height of G when the assumption ðjGj; jAjÞ 1⁄4 1 is replaced by the assumption that A is nilpotent (see [1]). In [9], Cheng Kei-Nah showed that G is metanilpotent if A is a cyclic group whose order is a product of two distinct primes. Here we obtain a result that takes Kei-Nah’s work one step further:

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

Abstract We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F,h]=F

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

{[ F,h] =F}

Action of a frobenius-like group with kernel having central derived subgroup

for all nonidentity elements h ∈ H. 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.

Fixed-point free action of an abelian group of odd non-squarefree exponent

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C C G (F)(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

The Influence of Hughes Type Action

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that [F,h] = F for all nonidentity elements h ∈ H. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with [F′,H] = 1. In case where CG(F) = 1 we prove that the groups G and CG(H) have the same nilpotent length under certain additional assumptions.

A GENERALIZED FIXED-POINT-FREE ACTION

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.

On finite groups admitting a special noncoprime action

We call the action of an automorphism α of a finite group G a Hughes type action if it is described by conditions on the orders of elements of G ⟨ α ⟩ − G. In the present paper we study the structure of finite group G admitting an automorphism α of prime order p so that the orders of elements in G ⟨ α ⟩ − G are not divisible by p 2.

Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms

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.

Fixed point free action on groups of odd order

An important result of Turull (1984) is the following: Let GA be a finite solvable group, G? GA and (|G|,|A|) = 1. Then f(G) < f(C G (A)) + 2l(A), where f denotes the Fitting height and l denotes the composition length. The purpose of this work is to give a treatment of the minimal configuration in this framework with additional conditions, yet without the coprimeness condition.