I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some subgroup $H\subseteq G$.
By a "conceptual" proof, I mean one that I can explain it to an undergraduate on a walk around the campus. I would like the proof to be from first principles, i.e., avoid relying on some other theorems about $p$-groups (e.g., Cauchy's theorem).
The standard way the lemma is proved is by finding an element $x\in G$ such that $\bar{a}\in G/\langle x\rangle$ is also of maximal order, and then using strong induction to decompose $G/\langle x\rangle$. However, I don't have a good conceptual explanation of how one finds the desired $x$. This is the only portion of the proof of structure theory for finite abelian groups for which I don't have a good "conceptual" reasoning.
