Definition 13

Let $G$ be a group with operators and $(H_i)_{i \in I}$ a family of normal stable subgroups of $G$. Let $p_i : G \to G / H_i$ be the canonical homomorphism. $G$ is called the internal product (or product) of the family of quotient groups $(G / H_i)$ if the homomorphism $g \mapsto (p_i(g))_{i \in I}$ is an isomorphism of $G$ onto $\prod_{i \in I} G / H_i$.

Let $G$ and $H$ be groups with operators and let $\phi$ and $\psi$ be two homomorphisms of $G$ into $H$. The set of elements $x$ in $G$ such that $\phi(x) = \psi(x)$ is a stable subgroup of $G$, called the coincidence group of $\phi$ and $\psi$. In particular, let $\phi_1 : G_1 \to H$ and $\phi_2 : G_2 \to H$ be homomorphisms of groups with operators; the coincidence group of the homomorphisms $\phi_1 \circ \mathrm{pr}_1$ and $\phi_2 \circ \mathrm{pr}_2$ of $G_1 \times G_2$ into $H$ is called the fibre product of $G_1$ and $G_2$ over $H$ relative to $\phi_1$ and $\phi_2$. It is denoted by $G_1 \times_H G_2$ when there is no ambiguity about $\phi_1$ and $\phi_2$ and the restrictions $p_1$ and $p_2$ of $\mathrm{pr}_1$ and $\mathrm{pr}_2$ to $G_1 \times_H G_2$ are also called projection homomorphisms. Then $\phi_1 \circ p_1 = \phi_2 \circ p_2$. The elements of $G_1 \times_H G_2$ are the ordered pairs $(g_1, g_2) \in G_1 \times G_2$ such that $\phi_1(g_1) = \phi_2(g_2)$. If $f_i$ is a homomorphism of a group with operators $K$ into $G_i$ ($i = 1, 2$) and $\phi_1 \circ f_1 = \phi_2 \circ f_2$, there exist one and only one homomorphism $f$ of $K$ into $G_1 \times_H G_2$ such that $f_i = p_i \circ f$ for $i = 1, 2$.