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.
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