Exercise 9, § 4

Let H be a connected closed subgroup of G, of maximum rank, distinct from G and maximal for these properties. Denote by Z the quotient group $C(H)/C(G)$.

a) We have dim $Z\leq 1$; if dim Z = 0, then Z is of order $2,3$ or 5 (reduce to the case in which G is almost simple, with trivial centre; apply Chap. VI, §4, Exerc. 4).

b) Assume that G is almost simple and that Z is of order 3 or 5; put $z=$ Card(Z) and $\pi =$ Card($\pi_1(H)$)$/$Card($\pi_1(G)$). Show that we must be in one of the following seven cases:

(i) G is of type $G_2, H$ is of type $A_2$, and $z= 3, \pi = 1$;

(ii) G is of type $F_4, H$ is of type $A_2\times A_2$, and $z=\pi = 3$;

(iii) G is of type $E_6, H$ is of type $A_2\times A_2\times A_2$, and $z=\pi = 3$; (iv) G is of type $E_7, H$ is of type $A_2\times A_5$, and $z=\pi = 3$;

(v) G is of type $E_8, H$ is of type $A_8$, and $z=\pi = 3$;

(vi) G is of type $E_8, H$ is of type $A_2\times E_6$, and $z=\pi = 3$;

(vii) G is of type $E_8, H$ is of type $A_4\times A_4$, and $z=\pi = 5$.

(Use loc. cit. and the plates of Chap. VI; to calculate $\pi$, remark that if $f$ and $f'$ denote the connection indices of G and H, respectively, then $z\pi f=f'.$) c) In each of the preceding cases, determine the group H.