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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