= Solution
For the <Abelian> case of <Cauchy theorem for groups>, use induction on $|G|$. Choose $a\ne e$. If $p$ divides $m=|a|$, then $a^{m/p}$ has order $p$. Otherwise $\langle a\rangle$ is a nontrivial <normal subgroup> of order prime to $p$, and $G/\langle a\rangle$ has smaller order still divisible by $p$. Induction gives a coset of order $p$; any lift $b$ has order divisible by $p$, because the order of its <image> divides its own order. A suitable power of $b$ then has order $p$. This uses no classification of finite <Abelian groups>.
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