SUBGROUPS OF MAXIMAL RANK IN FINITE EXCEPTIONAL GROUPS OF LIE TYPE
SUBGROUPS OF MAXIMAL RANK IN FINITE EXCEPTIONAL GROUPS OF LIE TYPE
复制标题
DOI：
10.1112/plms/s3-65.2.297
复制
发表时间：
1992-09
影响因子：
1.8
通讯作者：
M. Liebeck;J. Saxl;G. Seitz
中科院分区：
文献类型：
--
作者：
M. Liebeck;J. Saxl;G. Seitz
本文的目的是研究李型有限例外群的一大类自然极大子群，我们称之为极大秩子群。这些子群在[9，定理1]中的局部极大子群的分类中和在[28，定理2]中的子群的约化定理中都起着重要的作用：在[9]中证明了李型有限例外群的任何局部极大子群要么是极大秩子群，要么是一个小例外列表的子群；在[28]中，利用这个结果证明了任何极大子群要么是极大秩子群，要么是几乎单的，或者在已知的例外列表中。为了描述这些子群，我们需要以下符号。设G是特征为p>；0的代数闭域K上的简单伴随代数群，O是G的一个自同态，使得L=(Ga)‘是Fq上的有限李型例外单群，其中q=p.让A吧！“是使F*(X)=L的群。群Aut L由GA>生成；以及场和图的自同构(见[5，38])，所有这些都扩展到与o交换的抽象群G的态射。因此，Caut(C)(A)的一个子群X使得X=X/(O)，因此X作用在G的a-稳定子集的集合上。对于a-稳定子集Y，我们记为Y的X中的稳定子的NX(Y)。如果D是G的一个包含G的最大环面T的-稳定的闭约化子群，且M=NX(D)，我们称M为X中极大秩子群。本文确定了极大秩子群在X中极大的结构和共轭类。
The purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of Lie type, which we call subgroups of maximal rank. These subgroups play a prominent role both in the classification of local maximal subgroups in [9, Theorem 1] and in the reduction theorem for subgroups in [28, Theorem 2]: in [9] it is shown that any local maximal subgroup of a finite exceptional group of Lie type is either a subgroup of maximal rank or one of a small list of exceptions; and in [28], using this result, it is proved that any maximal subgroup is either of maximal rank, or almost simple, or in a known list. To describe these subgroups, we require the following notation. Let G be a simple adjoint algebraic group over an algebraically closed field K of characteristic p>0, and let o be an endomorphism of G such that L = (Ga)' is a finite exceptional simple group of Lie type over Fq, where q =p . Let A!" be a group such that F*(X) = L. The group Aut L is generated by Ga> together with field and graph automorphisms (see [5,38]), all of which extend to morphisms of the abstract group G commuting with o. Thus there is a subgroup X of CAut(C)(a) such that X = X/(o), and so X acts on the set of a-stable subsets of G. For a a-stable subset Y, we write NX(Y) for the stabilizer in X of Y. If D is a a-stable closed connected reductive subgroup of G containing a maximal torus T of G, and M = NX(D), we call M a subgroup of maximal rank in X. In this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in X.