Classification of some 3-subgroups of the finite groups of Lie type E6

Jianbei An, Heiko Dietrich, Shih Chang Huang

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

Original languageEnglish
Pages (from-to)4020-4039
Number of pages20
JournalJournal of Pure and Applied Algebra
Volume222
Issue number12
DOIs
Publication statusPublished - 1 Dec 2018

Cite this

An, Jianbei ; Dietrich, Heiko ; Huang, Shih Chang. / Classification of some 3-subgroups of the finite groups of Lie type E6. In: Journal of Pure and Applied Algebra. 2018 ; Vol. 222, No. 12. pp. 4020-4039.
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Classification of some 3-subgroups of the finite groups of Lie type E6. / An, Jianbei; Dietrich, Heiko; Huang, Shih Chang.

In: Journal of Pure and Applied Algebra, Vol. 222, No. 12, 01.12.2018, p. 4020-4039.

Research output: Contribution to journalArticleResearchpeer-review

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AB - We consider the finite exceptional group of Lie type G=E6 ε(q) (universal version) with 3|q−ε, where E6 +1(q)=E6(q) and E6 −1(q)=2E6(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.

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