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PRIMITIVE SUBGROUPS AND PST-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  18 July 2013

A. BALLESTER-BOLINCHES ,
J. C. BEIDLEMAN  and
R. ESTEBAN-ROMERO
A. BALLESTER-BOLINCHES
Affiliation:
Departament d’Àlgebra, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain email adolfo.ballester@uv.esresteban@uv.es
J. C. BEIDLEMAN*
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA
R. ESTEBAN-ROMERO
Affiliation:
Departament d’Àlgebra, Universitat de València, Dr. Moliner 50, 46100 Burjassot, València, Spain email adolfo.ballester@uv.esresteban@uv.es
*
✉clark@ms.uky.edu
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Abstract

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All groups considered in this paper are finite. A subgroup $H$ of a group $G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of $G$ containing $H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of $G$ has index a power of a prime if and only if $G/ \Phi (G)$ is a solvable PST-group. Let $\mathfrak{X}$ denote the class of groups $G$ all of whose primitive subgroups have prime power index. It is established here that a group $G$ is a solvable PST-group if and only if every subgroup of $G$ is an $\mathfrak{X}$-group.

Keywords

finite groupsprimitive subgroupssolvable PST-groupsT0-groups

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 373 - 378
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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