Presentation of direct product is disjoint union of presentations plus commutation relations

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups given by presentations. Then, the fact about::external direct product of $$G_1 \times G_2$$ can be given a presentation as follows: make disjoint the generating sets for $$G_1$$ and $$G_2$$ (i.e., make sure that different letters are used for generators). Then:


 * The generating set for $$G_1 \times G_2$$ is a disjoint union of the generating sets for $$G_1$$ and $$G_2$$.
 * The relation set for $$G_1 \times G_2$$ is as follows: The disjoint union of the relations for $$G_1$$, the relations of $$G_2$$, and the following commutation relations: a relation for every generator of $$G_1$$ and every generator of $$G_2$$ claiming that they commute.

For more than two groups
The same idea applies to more than two groups: make disjoint the generators, then take disjoint union of the generating sets and of the relations, and add a commutation relation for every pair of generators that arise from different factors.

When dealing with finitely many groups, this still gives the external direct product. However, when dealing with infinitely many groups, the group we get is not the external direct product but rather the restricted external direct product, which is the subgroup of the external direct product where only finitely many coordinates are allowed to be non-identity elements.

Related facts

 * Presentation of free product is disjoint union of presentations
 * Presentation of semidirect product is disjoint union of presentations plus action by conjugation relations