# Definition:Direct Limit of Sequence of Groups/Definition 1

## Definition

Let $\left({G_n}\right)_{n \in \N}$ be a sequence of groups.

For each $n \in \N$, let $g_n: G_n \to G_{n+1}$ be a group homomorphism.

A direct limit for the sequences $\left({G_n}\right)_{n \in \N}$ and $\left({g_n}\right)_{n \in \N}$ comprises:

a group $G_\infty$;
for each $n \in \N$, a group homomorphism $u_n: G_n \to G_\infty$;

such that, for all $n \in \N$:

$u_{n+1} \circ g_n = u_n$

and, for all groups $H$ together with group homomorphisms $h_n: G_n \to H$ satisfying $h_{n+1} \circ g_n = h_n$, there exists a unique group homomorphism:

$h_\infty: G_\infty \to H$

such that for all $n \in \N$:

$h_n = h_\infty \circ u_n$