elementary abelian subgroup of finite non-cyclic $p$-group

Question

Let $G$ be a finite non-cyclic $p$-group of order $p^n$, $n >1$, where $p$ is odd prime. I need to prove(By elementary methods)that $G$ has a subgroup isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p$?

Answer 1

If $G$ is abelian, then $G$ has at least two distinct subgroups of order $p$ such as $H,K$. Thus $H \times K \cong \Bbb{Z}_p \times \Bbb{Z}_p $.

Now let $G$ is non-abelian. We use induction on $n$. If all maximal subgroups of $G$ are cyclic, then $G \cong Q_8$, a contradiction. Thus there exists a non-cyclic maximal subgroup of $G$ such as $H$. By induction hypothesis $H$ has a non-cyclic subgroup of order $p^2$. Thus $G$ has a non-cyclic subgroup of order $p^2$.