Let $(\mathbb{N},\leq$) be a directed set where $m\leq n$ if an only if $m$ divides $n$.
We define a directed system of groups where $G_{n}=\mathbb{Z}$ for all $n\in \mathbb{N}$ and $f_{mn}\colon \mathbb{Z} \to \mathbb{Z}$ is multiplication by $\frac{n}{m}$.
I am trying to compute the directed limit of that system. The definition that I know is $$\dot{\bigcup}_{n\in \mathbb{N}} G_{n} / \sim,$$ where $z \sim y$ if $f_{mn}(z)=f_{kn}(y)$ where $n$ divides $m$ and $n$ divides $k$.
What I have tried is the following: in $G_{2}=\mathbb{Z}$, $2n=n$ where $n\in G_{1}$. In $G_{3}=\mathbb{Z}$, $3n=n$ where $n\in G_{1}$. In $G_{4}$, $4n=n=2n$ where $n\in G_{1}$ and $2n\in G_{2}$. In general, in $G_{k}$, $kn=n=k_{2}n=k_{3}n=\cdots=k_{m}n$ where $k_{2},\cdots,k_{m}$ are the divisors of $k$, $n\in G_{1}$ and $k_{i}n\in G_{i}$. Nevertheless, I do not know how to continue.
Can someone help me, please? Thank you in advance!
$$\underrightarrow{\lim}G_n\simeq \mathbf{Q},$$ the additive group of rational numbers.
Indeed, denote by $|$ divisibility in $\mathbf{N}_{>0}$. Since $\{n!:n\ge 0\}$ is cofinal in $(\mathbf{N}_{>0},|)$, the direct limit is the same as the direct limit restricted to this cofinal sequence $H_n=G_{n!}=\mathbf{Z}$ with given map $H_{n-1}\to H_{n}$ given by multiplication by $n$.
It is equivalent to write $H'_n=\frac{1}{n!}$ and consider the direct limit using identity maps. This makes clear that the direct limit is the union of this increasing sequence of subgroups of $\mathbf{Q}$, namely $\mathbf{Q}$ itself.