Proposition 11

Let $(\phi_i : H_i \to G_i)_{i \in I}$ be a family of homomorphisms of groups with operators and let $\Phi = \prod_{i \in I} \phi_i$. Then:

(a) $\mathrm{Ker}(\Phi) = \prod_{i \in I} \mathrm{Ker}(\phi_i)$; in particular, if all the $\phi_i$ *are injective, $\Phi$ is injective.

(b) $\mathrm{Im}(\Phi) = \prod_{i \in I} \mathrm{Im}(\phi_i)$; in particular, if all the $\phi_i$ *are surjective, $\Phi$ is surjective.

This is immediate.

In particular, let $(G_i)_{i \in I}$ be a family of groups with operators and, for all $i$, let $H_i$ be a stable (resp. normal stable) subgroup of $G_i$. The product $\prod_{i \in I} H_i$ is a stable (resp. normal stable) subgroup of $\prod_{i \in I} G_i$ and the canonical mapping of $\prod_{i \in I} G_i$ onto $\prod_{i \in I} (G_i / H_i)$ defines when passing to the quotient an isomorphism of

onto $\prod_{i \in I} (G_i / H_i)$. For example, let $J$ be a subset of $I$. The subgroup $G_J$ of $\prod_{i \in I} G_i$ consisting of the $(x_i)_{i \in I}$ such that $x_i = e_i$ for $i \notin J$ is a normal stable subgroup. The mapping $\iota_J$ which associates with $x = (x_j)_{j \in J}$ the element $y = (y_i)_{i \in I}$ such that $y_i = e_i$ for $i \notin J$ and $y_i = x_i$ for $i \in J$, is an isomorphism of $\prod_{j \in J} G_j$ onto $G_J$. The mapping $\mathrm{pr}_{I-J}$ defines when passing to the quotient an isomorphism $\theta_J$ of the quotient group $G / G_J$ onto $\prod_{i \in I-J} G_i$. The composition $\mathrm{pr}_J \circ \iota_J$ is the identity mapping of $\prod_{j \in J} G_j$. $G / G_J$ is often identified with $\prod_{i \in I-J} G_i$ because of $\theta_J$ and $\prod_{i \in J} G_i$ with $G_J$ because of $\iota_J$.

If $J_1$ and $J_2$ are disjoint subsets of $I$, it follows from the definitions that every element of $G_{J_1}$ commutes with every element of $G_{J_2}$.