EXCEPTIONAL ALGEBRAIC GROUPS 3 To deduce this from Theorem 1, observe that by [BT, 3.12], UX lies in a parabolic subgroup QL of G with U Q then any two closed complements to U in UX are also closed complements to Q in QX, so Theorem 1 ensures that these complements are Q-conjugate. Taken together with [Se2, Theorem 1], Theorem 1 leads to a description of all closed semisimple subgroups of G (see Theorems 5 and 7 below). The next result is also a consequence of Theorem 1. Theorem 2 Let X be a closed connected reductive subgroup of G, and assume that p = 0orp N(X,G). Then (i) CG(X)° is reductive (ii) if p 0 then OP(CG(X)) 1 (where OP(CG(X)) denotes the largest normal p-subgroup of CG(X)) (Hi) if X is semisimple, then the rank of CG(X) is equal to the maximal co-rank among subsystem subgroups of G containing X. We also establish a result on centralizers of non-connected reductive subgroups (see Corollary 4.5). The determination of all closed simple subgroups of G leads to the next result. Theorem 3 Let X be a simple closed connected subgroup of G with rank(X) 2, and assume that either p = 0 or p is a good prime for G and p N(Xy G). Then CL(G)(X) = L(CG(X)). Remarks 1. The conclusion of Theorem 3 has been shown to hold also for X = A\ in [LT]. 2. The analogues of Theorems 2 and 3 for classical groups are not in general true. For example, if G SL(V) and X is a subgroup of G such that V is indecomposable for X with composition series 0 V\ V2 V, where V\ = V/V2, then CG(X) is not reductive. And if G SLn with p = n, then CL^G){G) / L(CG(G)). The next theorem and its corollary concern the connection between (Aut G)- conjugacy and linear equivalence on L(G) for subgroups of G. Theorem 4 Let X\ and X2 be closed connected simple subgroups of G of the same type, and assume that p 0 or p N(Xi,G). Suppose that X\ and X2 have the same composition factors on L(G) (counting multiplicities). Then either X\ is conjugate to X2 in Aut G, or G E8 and Xi = X2 = A2, with both X\ and X2 lying in subsystem groups D4D4 and projecting irreducibly in each factor.
Previous Page Next Page