Graph product of groups

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: (facts closely related to Graph product of groups, all facts related to Graph product of groups) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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.