Let G be a group with $|G|=p^{2}$ , prove that G is cyclic.

Question

We're given a group with order $p^{2}$ ,where $p$ is a prime, we need to show that $G$ is cyclic.

Since its order is $p^{2}$ there's an element $a$ $\in$ $G$ such that $a^{p^{2}}= 1$ ,

Could anyone tell how to proceed from here or any other approach?

Answer 1

Let $G$ be the group $C_2\times C_2$ of order $p^2=2^2=4$. It has no element of order $4$, and hence it is not cyclic.

What is true is that every group of order $p^2$ with $p$ prime is abelian. This has been shown here several times, e.g., here.