Amalgamated free product

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

Definition

Definition with strict common subgroup

Let $G_{1}$ and $G_{2}$ be two groups, and let $H$ be a group with an injective homomorphism to both $G_{1}$ and $G_{2}$. Then the amalgamated free product of $G_{1}$ and $G_{2}$ via $H$ is defined as the quotient of the free product of $G_{1}$ and $G_{2}$, by the relation of the $H$ in $G_{1}$ being the same as the $H$ in $G_{2}$.

Definition of weaker notion (pushout version)

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