Where can we find a well developed material on direct limits of finite $p$-groups?


For instance, is there a characterization of such groups, which have a finite rank (that is every subgroup can be generated by a finite positive integer $k$?


Or, is it true that such a group is finite, provided that it is finitely generated and periodic (finite exponent)?


I have no idea about how much these questions are appropriate for this site, so I ask for your excuse.



A question on direct limits of finite $p$-groups
(Too long for a comment) A finitely generated subgroup of a direct limit of groups is isomorphic to a quotient of a subgroup of one of the constituent groups.


Indeed, suppose that $\{G\_i, f\_{ij}\}\_{i\in I}$ is a directed family, $G$ a subgroup of $\lim\limits\_{\rightarrow} G\_i$, and that $G=\langle g\_1,\ldots,g\_n\rangle$. The elements of $G$ are equivalence classes of pairs $(x,i)$, with $i\in I$ and $x\in G\_i$, where $(x,i)\sim (y,j)$ if and only if there exists $k\geq i,j$ such that $f\_{ik}(x)=f\_{jk}(y)$. For each $i=1,\ldots,n$, there exists $j\_i\in I$ such that $g\_i=[x\_i,j\_i]$. Let $k\geq j\_1,\ldots,j\_n$. Then $g\_i = [f\_{j\_ik}(x\_i),k]$, so that $g\_1,\ldots,g\_n$ can be identified with elements of $G\_k$; hence, $G$ is isomorphic to the image of this subgroup inside $G$ (via the canonical map from $G\_k$ into $G$).


So if all the $G\_k$ are finite, then so is $G$. If all the $G\_k$ are finite $p$-groups, then so is $G$. The class of groups that are finite rank and direct limits of finite $p$-groups are exactly the finite $p$-groups. 


