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Finite simple groups which projectively embed in an exceptional Lie group are classified! (English) Zbl 0916.22008
This is a survey paper describing the classification, up to isomorphism, of the finite simple groups which are subquotients of \(E_8 (\mathbb{C})\). As well as a table of results, there are descriptions of the main techniques used, and an extensive (but by no means exhaustive) list of references. The basic tools of character theory and local analysis are used to reduce to a manageable list of cases, which are then treated individually. The “hard” cases have all been resolved by computer calculations in \(E_8(F)\) for suitable finite fields \(F\). It is worth mentioning that the analogous problem for subquotients of \(E_8(F)\), where \(F\) is a finite field, is also solved. As in the present case this is the work of many people. A full description can be found in work of P. B. Kleidman and the reviewer [J. Algebra 157, No. 2, 316-330 (1993; Zbl 0794.20024)] for the embedding of sporadic groups, and M. W. Liebeck and G. M. Seitz [On finite subgroups of exceptional algebraic groups (to appear)] for cross-characteristic embeddings of groups of Lie type.

MSC:
22E40 Discrete subgroups of Lie groups
20E28 Maximal subgroups
20D08 Simple groups: sporadic groups
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