Action of a Frobenius-like group

Date

2014-03-15

Author

Güloǧlu, Ismail Ş.
Ercan, Gülin

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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 for all nonidentity elements h is an element of 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.

Subject Keywords

Frobenius action, Automorphisms, Fixed points

URI

https://hdl.handle.net/11511/35091

Journal

Journal of Algebra

DOI

https://doi.org/10.1016/j.jalgebra.2014.01.005

Collections

Department of Mathematics, Article

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Action of a Frobenius-like group with kernel having central derived subgroup Ercan, Gülin (2016-09-01) 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 is an element of H. Suppose that a finite group G admits a Frobenius-like group of auto-morphisms FH of coprime order with [F', H] = 1. In case where C-G( F) = 1 we prove that the groups G and C-G( H) have the same nilpotent length under certain additional assumptions.
Action of a Frobenius-like group with fixed-point free kernel Ercan, Gülin (Walter de Gruyter GmbH, 2014-09-01) 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 for all nonidentity elements h is an element of 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 th...
On the influence of fixed point free nilpotent automorphism groups Ercan, Gülin (2017-12-01) 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 for all nonidentity elements . Let FH be a Frobenius-like group with complement H of prime order such that is of prime order. Suppose that FH acts on a finite group G by automorphisms where in such a way that In the present paper we prove that the Fitting series of coincides with the intersections of with the Fitting series of G, and the nilpotent length of G exceeds the...
Frobenius-like groups as groups of automorphisms Ercan, Gülin; Khukhro, Evgeny (2014-01-01) 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 C-G(H) and focusing mainly on bounds for the Fitting height and rela...
A GENERALIZED FIXED-POINT-FREE ACTION Güloğlu, İsmail Şuayip; Ercan, Gülin (2013-05-01) 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 epsilon C-G(A) of prime order or of order 4, every conjugate of x in G is also contained in C-G(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.

Citation Formats

I. Ş. Güloǧlu and G. Ercan, “Action of a Frobenius-like group,” Journal of Algebra, pp. 533–543, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/35091.