Exercise 5, § 6

Let F be a group whose centre is $\{e\}$ and A its automorphism group. F is identified with a subgroup of A by means of the homomorphism Int: $F \to A$; let $\Gamma = A/F$.

(a) Show that the extension $F \to A \to \Gamma$ is trivial only if $\Gamma = \{e\}$ (note that the centralizer of F in A is $\{e\}$).

(b) Suppose that $\Gamma = \{e\}$. Show that every extension

of a group G by the group F is trivial.