# 群与图讨论班 Seminars on Groups and Graphs

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# Elementary abelian subgroups of finite classical groups of Lie type

## Meizheng Fu

University of Auckland

Time: 14:00-16:00 (GMT+8), Saturday March 18th, 2023
Location: Zoom

Abstract: Many open conjectures in the representation theory of finite groups can be studied by reducing to quasi-simple groups. An important tool in this is $$p$$-radical subgroups and their local structures. A $$p$$-radical subgroup is a subgroup $$R$$ of $$G$$ such that $$R=O_p(N_G(R))$$, the largest normal $$p$$-subgroup of the normalizer. Call a subgroup of a finite group $$G$$ $$p$$-local if it is the normalizer of nontrivial $$p$$-subgroup of $$G$$. One can show that every $$p$$-radical subgroup $$R$$ of $$G$$ with $$O_p(G)\neq G$$ is radical in some maximal-proper $$p$$-local subgroup $$M$$ of $$G$$. And it can be shown that every $$M$$ of $$G$$ can be realized as the normalizer of an elementary abelian $$p$$-subgroup. Thus, to classify $$p$$-radical subgroups of a finite group $$G$$, one can first classify elementary abelian $$p$$-subgroups. To classify elementary abelian $$p$$-subgroups of finite groups, we first do the classification in linear algebraic groups $$G$$ and then use a one-to-one correspondence derived from Lang-Steinberg theorem to transfer the results to finite groups of Lie type $$G^F$$, the fixed point subgroup of $$G$$ of the Steinberg endomorphism $$F$$. This approach was used by An, Heiko and Litterick to classify the local structure of elementary abelian $$p$$-subgroups in finite exceptional groups of Lie type. I will report briefly on my work to solve the problem for classical groups.

Host: 丁兆宸 Zhaochen Ding

Slides