Suppose $G$ is a group such that $\operatorname{ord(G)} = pq$ where $p, q$ are prime. Let $H$ be a proper subgroup of $G.$ Then $\operatorname{ord(H)} \mid pq$ and so it's possible that $\operatorname{ord(H)} = p.$ If $h \neq e \in H$, then $\operatorname{ord(h)} \mid p$ and so $\operatorname{ord(h)} = p \ldots$
How do we know $p \mid \operatorname{ord(h)}$ or $\operatorname{ord(h)} \not < p$ so that $\operatorname{ord(h)} = p$?