Writing $p$-groups using $p$-adics

Question

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.

Answer 1

Yes. If $A$ is a finite abelian $p$-group, then $A$ is a finitely generated $\mathbf{Z}_p$-module. Assume $A$ is generated by $n$ elements as a $\mathbf{Z}_p$-module (equivalently as an abelian group), say $a_1,\ldots,a_n$. Then we get a surjection $\pi:e_i\mapsto a_i:\mathbf{Z}_p^n\rightarrow A$ ($e_i$ being the standard basis vectors). The kernel $K$ of $\pi$ is a $\mathbf{Z}_p$-submodule of $\mathbf{Z}_p^n$, and since $\mathbf{Z}_p^n/K$ is finite, $K$ is (necessarily free) of rank $n$ over $\mathbf{Z}_p$. Let $x_1,\ldots,x_n$ be a basis for $K$, and let $A$ be the $n\times n$ matrix over $\mathbf{Z}_p$ such that $Ae_i=x_i$ for $1\leq i\leq n$. Then, viewing $A$ as the linear map $x\mapsto Ax$, we have $\mathrm{im}(A)=K$, so $A\cong\mathbf{Z}_p^n/K=\mathbf{Z}_p^n/\mathrm{im}(A)$.