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

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For two groups

Suppose G_1 and G_2 are groups given by presentations. Then, the 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.

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