Amalgamated free product

The amalgameted free product is a fibre coproduct along monomorphisms. Let be groups and and be monomorphisms. The amalgamated free product is the limit of the diagram

thus for any for which the diagram commutes, there exists a unique so that the diagram commutes:

If is the free product of and with inclusions and then the amalgamated free product is given by the quotient by a Normal closure: #m/thm/group

Proof

Let be the Normal closure

And be the quotient group with the projection . Let be the coproduct with injections and . Let such that the above diagram commutes. By the universal property of the coproduct, there exists a unique such that and . Hence and thus for all , implying . Then by the universal property of the quotient group, there exists a unique such that , and thus following diagram commutes:

Thus satisfies the universal property of the fibre product.

The above is a special case of the Fibre coproduct is the coëqualizer of a coproduct.

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