Free product with amalgamation

In abstract algebra, the free product of groups constructs a group from two or more given ones. Given, for example, groups G and H, the free product G*H can be constructed as follows: given presentations of G and of H, take the generators of G and of H, take the disjoint union of those, and adjoin the corresponding relations for G and for H. This is a presentation of G*H, the point being that there should be no interaction between G and H in the free product. If G and H are infinite cyclic groups, for example, G*H is a free group on two generators.
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