Theorem: A group $G$ of order $p^n$ is cyclic iff it is abelian having a unique subgroup of order $p$.
My problem is with the converse. If $a\in G$ have the largest order, say $p^k$, and $⟨a⟩$ is a proper subgroup of $G$, then there is $x\in G$ with $x\notin ⟨a⟩$ but with $x^p\in ⟨a⟩$.
I understand we must have an element of $G$ not in $⟨a⟩$ given that it's a proper subgroup. But I don't know why we must have such a $x$ so that $x^p$ is in $⟨a⟩$.
What I know is that $⟨G\setminus ⟨a⟩⟩ = G$, but don't know how this implies that we should have $x^p\in ⟨a⟩$ for a certain $x\notin ⟨a⟩$. Thanks in advance!