Proposition 2

Let $\mathscr{E}: F \xrightarrow{i} E \xrightarrow{p} G$ be an extension of $G$ by $F$. The following conditions are equivalent:

(i) $\mathscr{E}$ is a trivial extension; (ii) $\mathscr{E}$ has a retraction $r$; (iii) $\mathscr{E}$ has a section $s$ such that $s(G)$ is contained in the centralizer of $i(F)$.

Clearly (i) implies (ii) and (iii). If (ii) holds, the mapping $(r, p): E \to F \times G$ is a morphism of $\mathscr{E}$ into $\mathscr{E}_0$, whence (i). If (iii) holds, the homomorphism of $F \times G$ into $E$ corresponding to $(i, s)$ ($\S$ 4, no. 9, Proposition 12) is a morphism of $\mathscr{E}_0$ into $\mathscr{E}$, whence (i).

It may be that an extension $\mathscr{E}: F \to E \to G$ is not trivial and yet the group $E$ is isomorphic to $F \times G$ (Exercise 6).