When p groups are cyclic?

Question

Let $|G|=p^n$ and have only one subgroup of order $p^{(n-1)}$.Then G is cyclic.I am trying it in many ways bt get nothing. What I get :The unique subgroup is normal in $G$, Center meets the subgroup non trivially, No element outside the sub group has order $p^{(n-1)}$.Plz help me

Answer 1

Apparently this comes down to showing that every maximal subgroup of a finite $p$-group has index $p$. We do this inductively. Take $M\leq G$ maximal. Then there are two cases:

If $Z(G)\cap M = 1$, then $M$ and $Z(G)$ generate $G$ since $Z(G)\neq 1$. But $Z(G), M \leq N(M)$, so $G=N(M)$, i.e. $M$ is normal.

If $x\in Z(G)\cap M$, $x\neq 1$, then $M/(x) \leq G/(x)$ is a maximal subgroup. By induction, it is normal, so $M$ is normal.