Exercise 5, § 5

Assume that the group G is almost simple; let H be a principal connected closed subgroup of G (§4, Exerc. 18). Denote by $\Phi$ the group of automorphisms $u$ of G that fix H, and by F the subgroup of elements of G fixed by $\Phi$. a) Show that there exists a framing of G stable under $\Phi$.

b) Show that we must be in one of the following situations:

(i) $H = F_0$;

(ii) G is of type $B_3, H$ of type $G_2$ and $\Phi$ reduces to the identity;

(iii) G is of type $A_6, H$ of type $G_2$ and $\Phi$ of order 2.

(Use Exerc. 4 and Exerc. 20 of §4.)