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Let G be a simple algebraic group of exceptional type over an algebraically closed field K of characteristic p. The purpose of this paper is to determine all maximal closed subgroups of G which act irreducibly on either the ad-joint G-module or one of the well-known "minimal" modules of dimension 56, 27, 26-dp,3 or 7-dp,2 for G of type E7, E6, F4 or G2 respectively. (The adjective "minimal" here refers to the minimality of the dimension. ) This greatly extends [21, Theorem 4] for exceptional groups. A typical application of this sort of result is the following. We are given a finite subgroup X of G and from given information one determines a subgroup Y < X and a subgroup, Y < G of positive dimension such that Y and 炉 Y leave invariant precisely the same subspaces of V. If W is any X-invariant subspace of V, we conclude that W is stabilized by hX,, Y i, a group of positive dimension containing X. The maximal subgroups of positive dimension in G are known explicitly, so this yields information on the embedding of X in G, except when X acts irreducibly on V. Hence a list of irreducible subgroups is necessary for understanding the subgroups of G. Our result has already been applied in this way in [24]. In some small characteristics the adjoint module can be reducible for G (see [21, 1.10]): namely, it has a composition factor of codimension 1 for (G, p) = (E7, 2) or (E6, 3), and is also reducible for (G, p) = (F4, 2) or (G2, 3) (two composition factors of dimension 26, 26 or 7, 7 respectively). We write Vadj for the nontrivial composition factor of the adjoint module for G, excluding (G, p) = (F4, 2) or (G2, 3). And write Vmin for one of the 1

Subgroups of exceptional algebraic groups which are irreducible on an adjoint or minimal module