Graph product of groups

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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.
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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_vs with respect to the graph X is defined as F/R where F is the free product of all the G_vs 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.