A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive... AbstractWe show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a|G:H|3/2{a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most a⁢n3/2{an^{3/2}}maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is,the number of maximal subgroups of G containing H is at most |G:H|-1{\lvert G:H\rvert-1}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

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Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0222
Publisher site
See Article on Publisher Site

Abstract

AbstractWe show that there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a|G:H|3/2{a\lvert G:H\rvert^{3/2}}. In particular, a transitive permutation group of degree n has at most a⁢n3/2{an^{3/2}}maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stronger bound, that is,the number of maximal subgroups of G containing H is at most |G:H|-1{\lvert G:H\rvert-1}.

Journal

Forum Mathematicumde Gruyter

Published: May 1, 2020

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