16 MARTIN W. LIEBECK AND GARY M. SEITZ

Since A! is the unique SL2 in CXv(T{)°, we deduce that A' X n X

0

. Thus X n X

0

contains (ATi, A'), which is equal to X , proving the result in this case.

Now let X = G2,V = 10 ® 10^). Again let X

0

be a closed complement to V in

X 7 . Pick a fundamental subgroup A = A\ in X . The nontrivial composition factors

of i on F are 1 ® l^q\ 1 and 1^). Since p 3, none of these extends the trivial

module by 1.8, so as above we may assume that A X f l X

0

. Also

Cxv(A)0

— VbA',

where V0 - Cy{A) and A! - Cx(A) = A\. Moreover Vo is the irreducible A'-module

2 ® 2^), so again using 1.8 and 1.5, we may take AA! X 0 X0.

Let T2 be a maximal torus in AA!. Then Cxv(T

2

)° = ViT2, where Vi = Cy(T

2

) is

a 1-space; and there is an element w £ Nx(T2) of order 3 which centralizes V\. Thus

JVjfV^ ) contains a unique subgroup isomorphic to T2(w), from which it follows that

T2(w) X 0 Xo- Therefore X D X

0

contains (AA'^w), which is equal to X . This

completes the proof. •

Proposition 1.16 Let X Y be simple algebraic groups in characteristic p, with

( X , y , p ) = (2?

n

,I}

n

+i,2) (ra 2), (^4,^6,3) or (C3,As,3), m eac/i case ^ e natural

embedding. Let A = Ai, Ai or X2 in the respective cases, and V — Vy (A). Then there

is just one class of closed complements to V in XV.

Proof. In each case there is an X-composition series 0 V\ V2 V with V\ and

V/V2 both trivial 1-dimensional X-modules; moreover, V is indecomposable for X

(see [LS2,1.6]). Write W — V/V\, and suppose that there is more than one class ol

closed complements in XW. The proof of 1.5 implies that either there is a rational

indecomposable extension of W by the trivial X-module, or X = Cn and p — 2. The

latter is not the case, by hypothesis. Hence W extends the trivial module. Writing

V2/V\ — VxifJ*), 1.6 now implies that

Homx(rad(W;r(AA))?ii')

has dimension at least

2, which is a contradiction. •