Prove the order of each nonindentity element of $G$ is $p$.

Question

Sorry to disturbed, I have an abstract algebra coming and I'm unsure of how to prove this.

If $p$ is prime and $G$ is noncyclic group of order $p^2$, prove the order of each nonindentity element of $G$ is $p$.

Any help would be appreciated

Thank you in advance

Answer 1

Try thinking about it like this.

Every element of the group $G$ has order dividing $|G| = p^2$.

There are three positive integers dividing $p^2$. What are they?

How many elements have order $1$? Can any elements have order $p^2$?

What can you conclude about the orders of the remaining elements of $G$?