Subdirect product

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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