Exercise 4, § 5

Assume that G is semi-simple. Let $e$ be a framing of G and $\Phi$ a group of automorphisms of G respecting this framing; denote by H the subgroup of G consisting of the elements fixed by $\Phi$.

a) Show that $H_0$ is semi-simple; if G is simply-connected, H is connected (reduce to the case in which G is almost simple and $\Phi$ is cyclic, and apply Th. 1 (no. 3) and Chap. VIII, §5, Exerc. 13).

b) Assume that G is almost simple. Show that

if G is of type $A_{2l}(l\geq 1)$ and $\Phi$ is of order $2, H_0$ is of type $B_l$;

if G is of type $A_{2l-1}(l\geq 2)$ and $\Phi$ is of order $2, H_0$ is of type $C_l$;

if G is of type $D_l(l\geq 4)$ and $\Phi$ is of order $2, H_0$ is of type $B_{l-1}$;

if G is of type $D_4$ and $\Phi$ is of order 3 or $6, H_0$ is of type $G_2$;

if G is of type $E_6$ and $\Phi$ is of order $2, H_0$ is of type $F_4$.

In each case, determine the groups $\pi_i$(H), $i= 0,1$ (cf. Exerc. 3).

c) Prove that the subgroup $H_0$ is principal (§4, Exerc. 18).

d) Prove that a semi-simple group of type $B_3$ or $A_6$ contains a principal subgroup of type $G_2$ (use $b$) and Exerc. 20 of §4).