Finite simple subgroups of semisimple complex Lie groups – A survey. (English) Zbl 0849.20010

Cohen, Arjeh M.; Wales, David B.

Kantor, William M. (ed.) et al., Groups of Lie type and their geometries. Proceedings of the conference held in Como, Italy, June 14-19, 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 207, 77-96 (1995).

The authors survey recent results regarding embeddings of the covering groups of finite simple groups in complex Lie groups, especially the Lie groups of exceptional type. The universal covers of the simple connected complex Lie groups of exceptional type form a chain with respect to group embeddings: \[ G_2(C) < F_4(C) < 3.E_6(C) < 2.E_7(C) < E_8(C). \] In \(\S 4\) the authors indicate what is known about the occurrence of finite simple groups in each of these. In §2 they deal with some general theory, and in §3 with the connection between group embeddings in groups of Lie type defined over the complex numbers and those over a finite field. §5 deals with related embedding problems, mainly focussing on finite maximal subgroups of the same overgroups, and finite simple subgroups of simple algebraic groups in positive characteristic. §6 is concerned with an overview of the calculations needed to establish the harder embeddings. In the last §7, the computational aspects of the constructive proofs are discussed.
For the entire collection see [Zbl 0830.00029].

Reviewer: A.Kondrat’ev (Ekaterinburg)

MSC:

20D06	Simple groups: alternating groups and groups of Lie type
20G20	Linear algebraic groups over the reals, the complexes, the quaternions
20E07	Subgroup theorems; subgroup growth
22E10	General properties and structure of complex Lie groups

Keywords:

covering groups; finite simple groups; universal covers; simple connected complex Lie groups of exceptional type; group embeddings; maximal subgroups

Software:

LiE