JOURNAL OF ALGEBRA 46, 355-388 (1977) Subgroups of the Group E6(q) Which Are Generated by Root Subgroups BRUCE COOPERSTEIN Department of Mathematics, University of California at Santa Cruz, Santa Cruz, California 95064 Communicated by Marshall Hall, jr. Received November 19, 1975 1. INTRODUCTION Suppose G z G,(q) is a group of type w defined over the field GF(q). Then there is a class of subgroups, X, called root subgroups which generate G; each has order q and is parameterized by the field. Furthermore, the classesof groups in {(x, y): x, y E X} is rather small. In [3], McLaughlin considered the irreducible subgroups B of a group A,(q) z SL,+,(q) which are generated by root subgroups, where q # 2, and showed B = A,(q) or n is odd and B z Cta+I),z(q) z SP,+,(q). In [4], Stark determined those subgroups B of an orthogonal group S,(q) generated by root subgroups for the long roots such that B is transitive on the singular points of the associated orthogonal space. In this paper we consider a similar problem for the groups of type E,(q). More precisely we prove: THEOREM 1. Let G s E,(q), q = pa, p >, 5. Let X be the class of root sub- groups. Suppose B is a subgroup of G generated by the root subgroups it contains. If B is transitive in any parabolic representation of G, then B = G. 2. THE GROUP E,(q) AND ITS PARABOLIC REPRESENTATIONS Let G s E,(q). Let 52be the underlying root system and n = {CQ : 1 < i < 6} be a fundamental base for 52, the ordering chosen so that the Dynkin diagram for 52 is .- .-.- .-. . For ~EQ Copyright 0 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. ISSN 0021-8693