# Amalgamated free product

From Groupprops

This article describes a product notion for groups. See other related product notions for groups.

## Definition

### Definition with strict common subgroup

Let and be two groups, and let be a group with an injective homomorphism to both and . Then the amalgamated free product of and via is defined as the quotient of the free product of and , by the relation of the in being the same as the in .

### Definition of weaker notion (pushout version)

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