Existence and Uniqueness of Direct Limit of Sequence of Groups

== Theorem ==
Let $\sequence {G_n}_{n \mathop \in \N}$ be a sequence of groups.
Let $\sequence {g_n}_{n \mathop \in \N}: g_n: G_n \to G_{n + 1}$ be a sequence of group homomorphisms.
Then their direct limit $G_\infty$ exists and is unique up to unique isomorphism.
== Proof ==
=== Existence ===
Define $\widehat G_\infty$ by:
:$\widehat G_\infty := \ds \bigsqcup_{n \mathop \in \N} G_n$
where $\ds \bigsqcup$ signifies a disjoint union.
For $m \ge n$, denote with $g_{n, m}$ the composition $g_{m - 1} \circ \cdots \circ g_n: G_n \to G_m$.
In particular, $g_{n, n}: G_n \to G_n$ is taken to be the identity mapping on $G_n$.
Now define an equivalence relation $\sim$ on $\hat G_\infty$ by putting:
:$\tuple {x_n, n} \sim \tuple {y_m, m} \iff \exists k \ge n, m: \map {g_{n, k} } {x_n} = \map {g_{m, k} } {y_m}$
which is established to be indeed an equivalence relation in Lemma 1.
Now define $G_\infty$ as the quotient set:
:$G_\infty := \widehat G_\infty / \sim$
We can equip $G_\infty$ with a group structure as follows:
Let $\eqclass {\tuple {x_n, n} } {}, \eqclass {\tuple {y_m, m} } {} \in G_\infty$, where $\eqclass {\tuple {x_n, n} } {}$ denotes the equivalence class of $\tuple{x_n, n}$.
Let $l:= \max \set {m, n}$.
We define a group operation on $G_\infty$ via 
:$\tuple {\eqclass {\tuple {x_n, n} } {}, \eqclass {\tuple {y_m, m} } {} } \longmapsto \eqclass {\tuple {\map {g_{n l} } {x_n} \map {g_{m l} } {y_m}, l} } {}$.
We shall also refer to this operation as multiplication.
A proof that this is indeed a well-defined group can be found in Lemma 2.
Now define $u_n: G_n \to G_\infty$ by:
:$\map {u_n} {x_n} := \eqclass {\tuple {x_n, n} } {}$
That this is a group homomorphism follows immediately from the definition of the group structure.
Now let $H$ be a group and $h_n: G_n \to H$ be group homomorphisms such that $h_{n + 1} \circ g_n = h_n$.
Define $h_\infty: G_\infty \to H$ by:
:$\map {h_\infty} {\eqclass {\tuple {x_n, n} } {} } = \map {h_n} {x_n}$
Under this definition:
:$\forall x_n \in G_n: \map {h_\infty \circ u_n} {x_n} = \map {h_n} {x_n}$
By Equality of Mappings, it follows that:
:$h_\infty \circ u_n = h_n$
It remains to be shown that $h_\infty$ is well-defined and a group homomorphism.
This is done in Lemma 3.
=== Uniqueness ===
A direct proof of uniqueness can be formulated as follows:
Let $G_\infty'$ be a second direct limit group with homomorphisms $u_n': G_n \to G_\infty'$. 
By the above universal property we can construct homomorphisms:
:$u'_\infty: G_\infty \to G_\infty'$ (taking $H = G_\infty'$ and $h'_n = u'_n$)
and:
:$u_\infty: G_\infty' \to G_\infty$ (taking $H = G_\infty$ and $h_n = u_n$).
We shall prove that $u'_\infty$ is a left inverse of $u_\infty$.
Showing that it is also a right inverse follows analogously by exchanging $u_\infty$ and $u'_\infty$. 
From 
:$u'_\infty \circ u_n = u_n' = \mathop{Id}_{G'_\infty} \circ u'_n$
we find that the map $h_\infty : G'_\infty \to G'_\infty$ corresponding to $H = G'_\infty$ and $h_n = u'_\infty \circ u_n$ must be $\mathop{Id}_{G'_\infty}$ by uniqueness.
We also have
:$u'_\infty \circ u_n = u'_\infty \circ \mathop{Id}_{G_\infty} \circ u_n $
thus by the same argument $h_\infty$ must also equal $u_\infty'\circ u_\infty$. 
We conclude that $u_\infty'$ is a right inverse of $u_\infty$.
For a more categorical point of view:
From Equivalence of Definitions of Direct Limit of Sequence of Groups, $G_\infty$ can be described as a colimit in the category of groups $\mathbf{Grp}$.
By Colimit is Unique, it therefore is unique up to unique $\mathbf{Grp}$-isomorphism.
By Isomorphism in Category of Groups iff Group Isomorphism, the result follows.
== Sources ==
* : $\S 5.6$: Example $5.29$
Category:Group Theory
Category:Existence and Uniqueness of Direct Limit of Sequence of Groups