# Subdirect product

From Groupprops

## Definition

Let and be groups. A **subdirect product** of and is a subgroup of the external direct product such that the projection from to either direct factor is surjective. In other words, if is given by and is given by , then and .

## Examples

- The direct product is itself a subdirect product.
- In a direct product , the diagonal subgroup, given by is a subdirect product.
- More generally, if is a surjective homomorphism, the subgroup of given by is a subdirect product of and .