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6 results

  • Finite simple subgroups of semisimple complex Lie groups – a survey

  • Edited by William M. Kantor, University of Oregon, Lino Di Martino, Università degli Studi di Milano
  • Book:
    Groups of Lie Type and their Geometries
    Published online:
    06 January 2010
    Print publication:
    12 January 1995, pp 77-96

  • Summary

    Abstract

    We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type.

    Introduction

    Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F) and positive integers n. The topic of this survey is motivated by the question as to what happens if GL(n,·) is replaced by another algebraic group G(·).

    We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k. A further restriction of our discussion concerns the field k. It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G(ℂ). (See below for some exceptions in §3 and §5.)

    For G(·) of classical type, the theory for representations L → G(ℂ) differs little from the usual one for GL(n, ℂ). Indeed, a representation L → GL(n, ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

  • Finite Linear Groups of Degree Seven. I

  • David B. Wales
  • Journal:
    Canadian Journal of Mathematics / Volume 21 / 1969
    Published online by Cambridge University Press:
    20 November 2018, pp. 1042-1053
    Print publication:
    1969
    • Article
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  • 1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

  • Finite Linear Groups of Prime Degree

  • David B. Wales
  • Journal:
    Canadian Journal of Mathematics / Volume 21 / 1969
    Published online by Cambridge University Press:
    20 November 2018, pp. 1025-1041
    Print publication:
    1969
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  • If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.

    In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.