Theorem 3

Let $f : G \to G'$ be a homomorphism of groups with operators. (a) $\mathrm{Ker}(f)$ is a normal stable subgroup of $G$; (b) $\mathrm{Im}(f)$ is a stable subgroup of $G'$; (c) the mapping $f$ is compatible with the equivalence relation defined on $G$ by $\mathrm{Ker}(f)$; (d) the mapping $\tilde{f} : G/\mathrm{Ker}(f) \to \mathrm{Im}(f)$ derived from $f$ by passing to the quotient is an isomorphism of groups with operators; (e) $f = i \circ \tilde{f} \circ \pi$, where $i$ is the canonical injection of $\mathrm{Im}(f)$ into $G'$ and $\pi$ is the canonical homomorphism of $G$ onto $G/\mathrm{Ker}(f)$.

Assertion (b) follows from Proposition 6. The equivalence relation $f(x) = f(y)$ on $G$ is compatible with the group with operators structure on $G$. By Theorem 2 (no. 4), it is therefore of the form $y \in xH$, where $H$ is a normal stable subgroup of $G$ and $H$ is the class of the identity element, whence $H = \mathrm{Ker}(f)$. Assertions (a), (c) and (d) then follow. Assertion (e) is obvious (Set Theory, II, § 6, no. 5).