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SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS

Part of: Permutation groups Representation theory of groups

Published online by Cambridge University Press:  25 October 2016

PETER M. NEUMANN ,
CHERYL E. PRAEGER  and
SIMON M. SMITH
PETER M. NEUMANN*
Affiliation:
The Queen’s College, Oxford, OX1 4AW, UK email peter.neumann@queens.ox.ac.uk
CHERYL E. PRAEGER
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia email cheryl.praeger@uwa.edu.au
SIMON M. SMITH
Affiliation:
School of Mathematics and Physics, College of Science, University of Lincoln, Brayford Pool, Lincoln, LN6 7TS, UK email sismith@lincoln.ac.uk
*
✉peter.neumann@queens.ox.ac.uk
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Abstract

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This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Keywords

infinite permutation groupsfinite groupsmodulesmodular representations

MSC classification

Primary: 20B07: General theory for infinite groups
Secondary: 20C05: Group rings of finite groups and their modules 20C10: Integral representations of finite groups 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 1 , February 2017 , pp. 136 - 149
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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