Amalgamated free product

The amalgameted free product is a fibre coproduct along monomorphisms. Let $𝐺, 𝐻, 𝐾$ be groups and $𝜑 : 𝐾 ↣ 𝐺$ and $𝜓 : 𝐾 ↣ 𝐻$ be monomorphisms. The amalgamated free product $𝐺 ⨿ 𝐾 𝐻$ is the limit of the diagram

𝑍 𝜑 ↢ 𝐾 𝜓 ↣ 𝐻

thus for any $𝑄, 𝑗 1, 𝑗 2$ for which the diagram commutes, there exists a unique $ℎ$ so that the diagram commutes:

If $𝐺 ⨿ 𝐻$ is the free product of $𝐺$ and $𝐻$ with inclusions $𝜄 1 : 𝐺 \to 𝐺 ⨿ 𝐻$ and $𝜄 2 : 𝐻 \to 𝐺 ⨿ 𝐻$ then the amalgamated free product is given by the quotient by a Normal closure: group

𝐺 ⨿ 𝐻 / n c l {𝜄 1 𝜑 (𝑘) 𝜄 2 𝜓 (𝑘 - 1) : 𝑘 \in 𝐾}

Proof

Let $𝑁$ be the Normal closure
$𝑁 = n c l {𝜄 1 𝜑 (𝑘) 𝜄 2 𝜓 (𝑘 - 1) : 𝑘 \in 𝐾}$
And $𝐺 ⨿ 𝐻 / 𝑁$ be the quotient group with the projection $𝜋 : 𝐺 ⨿ 𝐻 ↠ 𝐺 ⨿ 𝐻 / 𝑁$ . Let $𝐺 ⨿ 𝐻$ be the coproduct with injections $𝜄 1 : 𝐺 \to 𝐺 ⨿ 𝐻$ and $𝜄 2 : 𝐻 \to 𝐺 ⨿ 𝐻$ . Let $𝑄, 𝑗 1, 𝑗 2$ such that the above diagram commutes. By the universal property of the coproduct, there exists a unique $𝑝 : 𝐺 ⨿ 𝐻 \to 𝑄$ such that $𝑝 𝜄 1 = 𝑖 1$ and $𝑝 𝑗 2 = 𝑗 2$ . Hence $𝑝 𝜄 1 𝜑 = 𝑝 𝜄 2 𝜓$ and thus $𝜄 1 𝜑 (𝑘) 𝜄 2 𝜓 (𝑘 - 1) \in k e r 𝑝$ for all $𝑘 \in 𝐾$ , implying $𝑁 \subseteq k e r 𝑝$ . Then by the universal property of the quotient group, there exists a unique $ℎ : 𝐺 ⨿ 𝐻 / 𝑁 \to 𝑄$ such that $ℎ 𝜋 = 𝑝$ , and thus following diagram commutes:

Thus $𝐺 ⨿ 𝐻 / 𝑁$ satisfies the universal property of the fibre product.

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Amalgamated free product

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