# Graph product of groups

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

## Contents

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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View a list of other standard non-basic definitions

## Definition

Let $X = (V,E)$ be a graph. To each $v$ in $V$, associate a vertex group $G_v$. Then, the graph product of the $G_v$s with respect to the graph $X$ is defined as $F/R$ where $F$ is the free product of all the $G_v$s and $R$ is the normal subgroup generated by subgroups of the form $[G_u,G_v]$ whenever there is an edge joining $u$ and $v$.

## Particular cases

### Direct product

The external direct product (restricted version) of a family of groups is precisely the same as the graph product with these as the vertex groups and the graph as the clique.

### Free product

The free product of a family of groups is precisely the same as the graph product with these as the vertex groups and the graph taken as the empty graph.