Find some nonisomorphic groups that are direct limits of cyclic groups of orders $p,p^2,p^3, \cdots$.
I know that the Prüfer group of type $p^{\infty}$ is a direct limit of cyclic groups of oders $p,p^2,p^3, \cdots$. In order to find other direct limits, I think the only thing I can do is to redefine the homomorphisms between the groups.
Suppose the groups $G_i$ is generated by $x_i$, of order $p^i$, $i=1,2, \cdots$. For any positive integers $i$ and $j$, if $i \leq j$, define the map $\rho_{ij}: G_i \rightarrow G_j$, $x_i \mapsto x_j^{p^{j-i}}$. Then $D=\{ G_i, \rho_{ij} : i \leq j, i,j \in \mathbb{Z}_+ \}$ is a direct limit of $G_i$, which is the Prüfer group. As $(p+1)$ is prime to $p$, if the new homomorphism $\rho_{ij}': G_i \rightarrow G_j$ maps $x_i$ to $x_j^{p^{j-i}(p+1)^{j-i}}$, it is also injective. Then another direct limit can be obtained.
Are the two direct limits $D=\{ G_i, \rho_{ij} : i \leq j, i,j \in \mathbb{Z}_+ \}$ and $D'=\{ G_i, \rho_{ij}' : i \leq j, i,j \in \mathbb{Z}_+ \}$ isomorphic? If they are not, how can I prove? [I think the key problem lies in determining the possible homomorphisms between $D$ and $D'$.]
Many thanks.