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

Ercan, Gülı̇n; Güloğlu, İsmaı̇l Ş.; Khukhro, E. I.

doi:10.1007/s10469-014-9287-4

Cited by 9 publications

(11 citation statements)

References 16 publications

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“…With this notation the above theorem leads to an analogue of Theorem 2.6 for Frobenius-like groups; namely we have the following result obtained in [4].…”

Section: The Rank Of P Is Bounded In Terms Of |H| and The Rank Of C Pmentioning

confidence: 83%

“…Namely, Theorem A in [2] is proved by reducing the structure of a minimal counterexample to a very restricted configuration and deducing a contradiction by proving the following theorem [2, Proposition C] on representations of some specific groups having a normal extraspecial subgroup, which is also of independent interest. We now consider the following complicated-looking hypothesis introduced in [4]. It is formulated to avoid the so-called exceptional cases, which possibly occur in Hall-Higman type arguments, and is slightly more general than assuming that F H is of odd order as in the hypothesis of Theorem A in [2].…”

Section: Frobenius-like Groupsmentioning

confidence: 99%

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Frobenius-like groups as groups of automorphisms

Ercan¹,

Güloğlu²,

Khukhro³

2014

Turk J Math

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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 FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms FH aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kerne

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“…With this notation the above theorem leads to an analogue of Theorem 2.6 for Frobenius-like groups; namely we have the following result obtained in [4].…”

Section: The Rank Of P Is Bounded In Terms Of |H| and The Rank Of C Pmentioning

confidence: 83%

Section: Frobenius-like Groupsmentioning

confidence: 99%

Frobenius-like groups as groups of automorphisms

Ercan¹,

Güloğlu²,

Khukhro³

2014

Turk J Math

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“…So far we have obtained several extensions of Khukhro's result replacing F H by a Frobenius-like group with kernel F under some mild additional conditions (see [3], [4], [5], [6], [7], [8], [9]). Unfortunately, this time we must be satisfied with the present form due to the example given in [4].…”

Section: Introductionmentioning

confidence: 99%

Frobenius action on Carter subgroups

Ercan

Güloğlu

2020

Int. J. Algebra Comput.

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“…For example, there are recent results obtained by Ercan, Güloğlu and Khukhro where the condition of BA being Frobenius was replaced by BA/ [B, B] being Frobenius, see for instance [3][4][5]. Earlier in [11], Shumyatsky also extended for dihedral groups some results proved for Frobenius groups.…”

Section: Introductionmentioning

confidence: 99%

On p-groups of maximal class as groups of automorphisms

Melo

2015

Journal of Algebra

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Let P be a p-group of maximal class and M be a maximal subgroup of P . Let α be an element in P \ M such that |C P (α)| = p 2 , and assume that |α| = p. Suppose that P acts on a finite group G in such a manner that C G (M ) = 1. We show that if C G (α) is nilpotent, then the Fitting height of G is at most two and C G (α) is contained in the Fitting subgroup of G. For p = 2, without assuming that C G (α) is nilpotent, we prove that the Fitting height h(G) of G is at most h(C G (α)) + 1 and the Fitting series of C G (α) coincides with the intersection of C G (α) with the Fitting series of G. It is also proved that if C G (x) is of exponent dividing e for all elements x ∈ P \ M , then the exponent of G is bounded solely in terms of e and |P |. These results are in parallel with known results on action of Frobenius and dihedral groups.

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

Cited by 9 publications

References 16 publications

Frobenius-like groups as groups of automorphisms

Frobenius-like groups as groups of automorphisms

Frobenius action on Carter subgroups

On p-groups of maximal class as groups of automorphisms

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