# Graph product of groups

From Groupprops

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

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Graph product of groups, all facts related to Graph product of groups) |Survey articles about this | Survey articles about definitions built on this

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## Definition

Let be a graph. To each in , associate a vertex group . Then, the graph product of the s with respect to the graph is defined as where is the free product of all the s and is the normal subgroup generated by subgroups of the form whenever there is an edge joining and .

## 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.