It is proved in this post that every group is the inductive limit of the family of its finitely generated subgroups (where the partial order is the inclusion). Indeed, given $G_i, G_j$ two finitely generated subgroups of $G$, the group generated by generators of $G_i, G_j$ is another finitely generated subgroup that contains $G_i, G_j$. Definitions of the inductive limit of groups can be found in the same post.
My question is: given $G$ a countably infinite group, when can $G$ be written as an inductive limit of finite subgroups; or equivalently, when can the inclusion, as a partial order, is directed in the family of finite subgroups? In general, given a countably infinite family of finite groups $\{G_i\}_{i\in\mathbb{N}}$, for each $N\in\mathbb{N}$, define:
$$
H_N = \prod_{1\leq i \leq N}G_i
$$
and, $\phi_{N, N+1}$, the connecting group homomorphism as follows:
$$
\phi_{N, N+1}: H_N\rightarrow H_{N+1}, \hspace{0.5cm} (g_1, \cdots, g_N) \mapsto (g_1, \cdots, g_N, e_{N+1})
$$
where $e_n$ denotes the identity of $G_n$ for each $n\in\mathbb{N}$. Then define:
$$
H = \prod_{i\in\mathbb{N}} G_i
$$
and:
$$
\Phi_N: H_N\rightarrow H, \hspace{0.5cm}(g_1, \cdots, g_N)\mapsto (g_1, \cdots, g_N, e_{N+1}, e_{N+2}, \cdots)
$$
Then $(H, \{\Phi_i\}_{i\in\mathbb{N}})$ is an inductive limit of finite subgroups where the group operation is defined coordinate-wise. I believe this is one way to build the desired group, or one situation when the inclusion is a partial order in the family of finite subgroups. Any concrete examples, or ideally a necessary condition for the statement to be true will be highly appreciated.
Best Answer
Inclusion is always a partial order of the family of finite subgroups. I think you mean to ask when this partial order is directed and covers $G$ (i.e. every element is contained in some finite subgroup), so that $G$ is a directed colimit (aka direct limit, aka inductive limit) of its finite subgroups.
The answer is: exactly when $G$ is a locally finite group, i.e. when every finitely generated subgroup is finite. Countability of $G$ is not relevant here.
If $G$ is locally finite, then since you already know that $G$ is a directed colimit of its finitely generated subgroups, and each such subgroup is actually finite, then $G$ is a directed colimit of its finite subgroups.
Conversely, suppose $G$ is a directed colimit of its finite subgroups. Let $g_1,\dots,g_n$ be finitely many elements of $G$. Each $g_i$ is in some finite subgroup $G_i$, and by directeness there is a finite subgroup $G_*$ containing all of the $G_i$. Now the subgroup of $G$ generated by $g_1,\dots,g_n$ is contained in $G_*$ and hence is finite. Thus every finitely generated subgroup of $G$ is finite, and $G$ is locally finite.
The wikipedia article on locally finite groups that I linked to above contains lots of examples, as well as closure properties of the class of locally finite groups, which you can use to generate many more examples.