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ON A GRAPH RELATED TO THE MAXIMAL SUBGROUPS OF A GROUP

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  14 January 2010

MARCEL HERZOG ,
PATRIZIA LONGOBARDI  and
MERCEDE MAJ
MARCEL HERZOG*
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, Israel (email: herzogm@post.tau.ac.il)
PATRIZIA LONGOBARDI
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy (email: plongobardi@unisa.it)
MERCEDE MAJ
Affiliation:
Dipartimento di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (Salerno), Italy (email: mmaj@unisa.it)
*
✉For correspondence; e-mail: herzogm@post.tau.ac.il
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Abstract

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Let G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

Keywords

maximal subgroupmaximal graphprime graphsolvable groupssimple groups

MSC classification

Secondary: 20E28: Maximal subgroups 20E32: Simple groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 317 - 328
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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