Presentation of free product is disjoint union of presentations

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For two groups

Suppose G_1 and G_2 are groups given by presentations. Then, a presentation for the external free product G_1 * G_2 is given as follows: we first make sure that all the letters for generators in the presentation of G_1 are different from the letters in the presentation of G_2 (i.e., we make disjoint the sets of generators). Now, we take the generating set for G_1 * G_2 as the union of both these generating sets and the relation set for G_1 * G_2 as the union of the relations for G_1 and G_2.

For a collection of more than two groups

The same rule applies: we make pairwise disjoint generating sets for all the groups, then take a disjoint union of the generators and of the relations.


In the examples, we use 1 to stand for the identity element. To avoid confusion, we already make disjoint the presentations for the two groups for which we are taking the free product.

G_1 Presentation G_2 Presentation G_1 * G_2 Presentation
group of integers \langle a \mid \rangle group of integers \langle b \mid \rangle free group:F2 \langle a,b \mid \rangle
cyclic group:Z2 \langle a \mid a^2 = 1 \rangle group of integers \langle b \mid \rangle  ? \langle a,b \mid a^2 = 1 \rangle
cyclic group:Z2 \langle a \mid a^2 = 1 \rangle cyclic group:Z2 \langle b \mid b^2 = 1 \rangle infinite dihedral group \langle a,b \mid a^2 = b^2 = 1 \rangle

More examples should be added to illustrate free product with more complicated presentations

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