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ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

Published online by Cambridge University Press:  22 August 2014

A. BALLESTER-BOLINCHES
Affiliation:
Departament d'Àlgebra, Universitat de València, 46100 Burjassot, València, Spain e-mail: Adolfo.Ballester@uv.es
J. C. BEIDLEMAN
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA e-mail: james.beidleman@uky.edu
A. D. FELDMAN
Affiliation:
Department of Mathematics, Franklin and Marshall College, Lancaster, PA 17604-3003, USA e-mail: afeldman@fandm.edu
M. F. RAGLAND
Affiliation:
Department of Mathematics, Auburn University Montgomery, Montgomery, AL 36124-4023, USA e-mail: mragland@aum.edu
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Abstract

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For a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

Keywords

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS
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