Maximal subgroups of finite simple groups: classifications and applications

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Abstract

This paper surveys what is currently known about the maximal subgroups of the finite simple groups. After briefly introducing the groups themselves, if their maximal subgroups are completely determined then we present this classification. For the remaining finite simple groups our current knowledge is only partial: we describe the state of play, as well as giving some results that apply more generally. We also direct the reader towards computational resources for the construction of maximal subgroups.

After this, we present three sample applications, selected because they combine group theoretical and combinatorial arguments, and because they use either or both of the detailed classifications and the looser statements that can be made about all maximal subgroups. In particu- lar, we discuss results relating to generation, and the generating graph; results concerning bases; and some applications to computational com- plexity, in particular to graph colouring and other problems with no known polynomial-time solution.
Original languageEnglish
Title of host publicationSurveys in Combinatorics 2021
EditorsKonrad K. Dabrowski, Maximillien Gadouleau, Nicholas Georgiou, Matthew Johnson, George B. Mertzios, Daniël Paulusma
PublisherCambridge University Press
Pages343-370
ISBN (Electronic) 9781009036214
DOIs
Publication statusPublished - 1 Jun 2021

Publication series

NameLondon Mathematical Society Lecture Note Series
PublisherCambridge University Press
Volume470
ISSN (Print)0076-0552

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