The authors consider the modular version of the problem of determining the finite subgroups of simple algebraic groups of exceptional types over the complex field. Let \(G\) be a simple algebraic group of type \(E_n\) (\(n=6,7,8\)), \(F_4\) or \(G_2\) over an algebraically closed field \(K\) of characteristic \(p\). The paper’s main theorem determines all finite simple groups \(S\) having a cover that imbeds into \(G\).
The investigation divides into two cases: \(S\in\text{Lie}(p)\) (the class of finite simple groups of Lie type) and \(S\notin\text{Lie}(p)\). The first case has been extensively studied and by now is well understood. For instance, in an earlier paper [Trans. Am. Math. Soc. 350, No. 9, 3409-3482 (1998;Zbl 0905.20031)] the authors showed that for \(S=S(q)\in\text{Lie}(p)\) and \(q\) sufficiently large, \(S\) is in a closed connected subgroup \(H\) of \(G\), where \(H\) is proper as long as \(S\) is not of the same type as \(G\).
The main result of the present paper contributes to the pursuit of the second case of the investigation: the authors determine the complete range of isomorphism types of finite simple subgroups of adjoint exceptional algebraic groups. These are listed in a set of four tables at the end of the paper. A single table gives a simpler version, which describes the possibilities for finite simple groups \(S\notin\text{Lie}(p)\) with some cover (that is, a perfect central extension) of \(S\) in a given exceptional algebraic group \(G\) of characteristic \(p\).
The approach first reduces the range of possibilities for subgroups \(S\) of exceptional groups to those in the set of tables referred to above. The authors then demonstrate that all such possibilities actually are realized: all such subgroups do occur in exceptional algebraic groups \(G\). The development relies on the work ofP. B. Kleidman andR. A. Wilson [J. Algebra. 157, No. 2, 316-330 (1993;Zbl 0794.20024)], which determined the isomorphism types of sporadic simple groups that lie in exceptional groups, as well as work ofJ.-P. Serre [Invent. Math. 124, No. 1-3, 525-562 (1996;Zbl 0877.20033)], which showed for a finite subgroup \(S\) of an exceptional simple complex Lie group with \(O_p(F)=1\) for a prime \(p\) that \(F\) is also a subgroup of \(G(\overline\mathbb{F}_p)\), where \(\overline\mathbb{F}_p\) is the algebraic closure of \(\mathbb{F}_p\).

Reviewer: J.F.Hurley (Storrs)