Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$. It has an answer here. $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$.
But when I was trying to prove it I have taken a map $f: \Bbb{Z}/p^k \Bbb{Z} \times A \to A$ s.t $(\bar n,a) \mapsto na$. This is bilinear over $\Bbb Z$ hence induces a map $F:\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A \to A$. Now my question is; is $Im(F)$ has order $p^k$. How to show that?