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Finite subgroups of Lie groups. (Sous-groupes finis des groupes de Lie.) (French) Zbl 0968.20025

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque 266, 415-430, Exp. No. 864 (2000).
This lecture gives, with the author’s usual conciseness and clarity, an account of the work of many authors on the classification of finite subgroups of a reductive group over \(\mathbb C\). This is classically equivalent to the study of finite subgroups of compact Lie groups, and can be seen as a generalization of the classical list of subgroups of the special orthogonal group \(\text{SO}_3(\mathbb R)\).
The following aspects are included: embeddings of finite Abelian groups and connection with cohomology, embeddings of finite simple groups (for instance, a necessary and sufficient condition is given for the existence of an embedding of \(\text{SL}_2(\mathbb F_q)\) in terms of the exponents of the corresponding Weyl group), projective embeddings of finite simple groups into exceptional groups.
For the entire collection see [Zbl 0939.00019].

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
22C05 Compact groups
22E40 Discrete subgroups of Lie groups
20E07 Subgroup theorems; subgroup growth
20D06 Simple groups: alternating groups and groups of Lie type
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