Litterick, Alastair J
(2018)
*On non-generic finite subgroups of exceptional algebraic groups.*
Memoirs of the American Mathematical Society, 253 (1207).
DOI https://doi.org/10.1090/memo/1207

Litterick, Alastair J
(2018)
*On non-generic finite subgroups of exceptional algebraic groups.*
Memoirs of the American Mathematical Society, 253 (1207).
DOI https://doi.org/10.1090/memo/1207

Litterick, Alastair J
(2018)
*On non-generic finite subgroups of exceptional algebraic groups.*
Memoirs of the American Mathematical Society, 253 (1207).
DOI https://doi.org/10.1090/memo/1207

## Abstract

The study of finite subgroups of a simple algebraic group G reduces in a sense to those which are almost simple. If an almost simple subgroup of G has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of G, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups Altn, Symn for n ≥ 10, do not occur as Lie primitive subgroups of an exceptional algebraic group. A subgroup of G is called G-completely reducible if, whenever it lies in a parabolic subgroup of G, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-G-completely reducible, non-generic simple subgroups. As an intermediate result, for each simply connected G of exceptional type, and each non-generic finite simple group H which embeds into G/Z(G), we derive a set of feasible characters, which restrict the possible composition factors of V ↓ S, whenever S is a subgroup of G with image H in G/Z(G), and V is either the Lie algebra of G or a non-trivial Weyl module for G of least dimension. This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for n ≥ 10, Altn and Symn, as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.

Item Type: | Article |
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Additional Information: | 158 pages; final version in Memoirs of the AMS. Minor edits with respect to the previous version |

Uncontrolled Keywords: | Algebraic groups; exceptional groups; finite simple groups; Lie primitive; subgroup structure; complete reducibility |

Divisions: | Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of |

SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |

Depositing User: | Unnamed user with email elements@essex.ac.uk |

Date Deposited: | 05 Sep 2019 10:25 |

Last Modified: | 18 Aug 2022 11:20 |

URI: | http://repository.essex.ac.uk/id/eprint/25253 |

## Available files

**Filename:** 1511.03356v2.pdf