Subdirect product

Let $G$ and $H$ be groups. A subdirect product of $G$ and $H$ is a subgroup $K$ of the external direct product $G \times H$ such that the projection from $K$ to either direct factor is surjective. In other words, if $p_1:G \times H \to G$ is given by $(g,h) \mapsto g$ and $p_2:G \times H \to H$ is given by $(g,h) \mapsto h$, then $p_1(K) = G$ and $p_2(K) = H$.
• In a direct product $G \times G$, the diagonal subgroup, given by $\{ (g,g) \mid g \in G \}$ is a subdirect product.
• More generally, if $\rho:G \to H$ is a surjective homomorphism, the subgroup of $G \times H$ given by $\{ g,\rho(g) \mid g \in G \}$ is a subdirect product of $G$ and $H$.