1. Introduction This work is a contribution to the study of the subgroup structure of simple algebraic groups defined over an algebraically closed field k. In the case where char k 0, the structure is described in detail in two papers of E.B. Dynkin [8], [9]. The case where char k is positive has been considered in [14], [15], [16], [17], [22], [23] and [28]. Interest revived in this subject after the classification of finite simple groups, and was generated by an attempt to understand the subgroup structure of the related finite simple groups of Lie type. We will concentrate here on the subgroup structure of exceptional algebraic groups in positive characteristic, where the most complete information available appears in [17] closed connected semisimple subgroups are investigated, under mild restrictions on char k. In the case where the subgroup is simple of rank at least 2, tables are presented listing all possibilities for the conjugacy class of the subgroup, its connected centralizer and its action on the Lie algebra of the overlying exceptional algebraic group. The present work seeks to provide similar detailed information for closed connected subgroups of type A\\ under mild restrictions on char k, we succeed in classifying all conjugacy classes of such subgroups, and present tables analogous to those of [17]. For brevity, we shall hereafter use the term "A\ subgroup" to mean "closed connected subgroup of type A{\ In addition, we take as motivation earlier work of the second author. In [31], a group-theoretic version of the Jacobson-Morozov theorem is established: if char k is a good prime for the simple algebraic group G, then any unipotent element of G having order p lies in an A\ subgroup. It is then natural to consider the question of how many nonconjugate A\ subgroups contain a given element of order p. We answer this question in the case where G is an exceptional group (again under mild restrictions on char k). We determine the unipotent class meeting each conjugacy class of A\ subgroups, and provide the information in the tables mentioned above. In particular we determine which elements of order p lie in a single conjugacy class of A\ subgroups of G. Throughout, unless otherwise stated G will be a simple algebraic group of exceptional type over an algebraically closed field k of characteristic p, and X a connected simple algebraic subgroup of G of rank 1. We write C(G) for the Lie algebra of G. Following [17], we shall assume that p N(Ai,G), where N(A1,G) is as given in the following table. G N{AUG) G2 3 F4 3 E6 5 E7 7 E8 7 Received by the editor November 13, 1995 and in revised form May 30, 1997. 1
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