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.
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