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