I have this exact sequence $$ 0\to \mathbb{Z}_{p^m}\overset{f}\to G\overset{g}\to \mathbb{Z}_{p^n}{\to} 0 $$ where $G$ is finitely generated. I want to prove that $|G|=p^{m+n}$ and a friend told me to look to $f(1_{\mathbb{Z}_{p^m}})$ and $g^{-1}(1_{\mathbb{Z}_{p^n}})$, where $1_{\mathbb{Z}_{p^m}}$ and $1_{\mathbb{Z}_{p^m}}$ are the generators of $\mathbb{Z}_{p^m}$ and $\mathbb{Z}_{p^m}$.
In fact, I see an image and preimage of generators are generators.
I've already proven that the image and the preimage are, indeed, generators. The main question is how to prove that they not generate the same group (I think that we use the fact of $im(f)=ker(g)$, but cannot concluded yet) and $f(1_{\mathbb{Z}_{p^m}})$ and $g^{-1}(1_{\mathbb{Z}_{p^n}})$ generate $G$.