Subdirect product

Definition
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 $$p_2(K) = H$$.

Examples

 * The direct product is itself a subdirect product.
 * 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 $$H$$.

Facts

 * Normal subdirect product of perfect groups equals direct product