EXCEPTIONAL ALGEBRAIC GROUPS

5

Theore m 5 Let X be a closed connected semisimple subgroup of G, such that every

simple factor of X has rank at least 2, and assume that p — 0 or p N(X,G).,

Choose a subsystem subgroup Y of G, minimal subject to containing X (possibly

Y — G of course). Then either

(i) X is essentially embedded in Y} or

(%%) Y — G — E$, p = 7, X — G2 and X F4 G with X maximal in F4.

The (Aut G)-conjugacy classes and connected centralizers of the simple subgroups

X (of rank at least 2) are given in Tables 8.1-8.5 in Section 8; also given are the mini-

mal subsystem subgroups containing X and the composition factors of the restrictions

L{G)[X.

Remark s (1) The essential embeddings in (i) can be completely determined using

results in Section 2 (indeed, we carry this out in justifying the information given in

Tables 8.1 - 8.5).

(2) For the convenience of the reader, there are two further tables, 8.6 and 8.7,

in Section 8. For G simply connected of type Ej or EQ, and X a simple subgroup

of rank at least 2, these tables give the restrictions V56 I X and V27 I X, where Vse

and V27 a r e the 56- and 27-dimensional modules

VE7(^7)

and

V E

6

( A I ) ,

respectively.

The next result concerns subgroups A\ of G. In Section 6 we define a labelled

diagram associated to any subgroup A\ of G; this is the Dynkin diagram of G, with

nodes labelled by various non-negative integers which are determined by the set of

weights on L(G) of a maximal torus of the subgroup.

Theore m 6 Assume that p — 0 or p N(A\,G). Then any subgroup A\ of G is

determined up to conjugacy in Aut G by its labelled diagram.

We shall see in Section 6 that subgroups of type A\ with the same composition

factors on L(G) have the same labelled diagram (see Theorem 6.3). Thus Theorem

6 is closely related to Theorem 4 in the A\ case.

Theorem 6 is proved using the next result, which is the analogue of Theorem 5

for subgroups with a factor A\.

Theore m 7 Let X be a closed connected semisimple subgroup of G with a factor

A\, and assume that p — 0 or p N{A\,G). Then one of the following holds:

(i) there is a subsystem subgroup Y of G containing X, such that Y is a product

of classical groups and X is essentially embedded in Y;

(ii) there is a subgroup Y0 — F^E^E-j or Eg of G, and a semisimple subgroup

Yi of

CG(YQ),

such that either

(a)X = Y0Yl, or

(b) X is essentially embedded in ZY\, where Z is a maximal connected

subgroup of Y$ not containing a maximal torus;

(in) G — G2, X = A\ and X is maximal in G.