Recently I was given a handout containing (roughly) the following text:
Let $A$ be a finite abelian group, and $p^k$ a prime power. The $p^k$-rank of $A$ is defined to be $\operatorname{rank}_{p^k}(A):=\operatorname{dim}_{\mathbb{F}_p}(p^{k-1}A/p^kA).$ It is immediate from the definition that for any subgroup $B\subseteq A$ and any prime power $p^k$ $\operatorname{rank}_{p^k}(B)\leq\operatorname{rank}_{p^k}(A).$
However, I do not see how this follows from the definition. Any suggestions?