Are any two groups of order 23 isomorphic to each other?

Question

I have to decide whether the following statements are true or false, with proofs.

1. Any two abelian groups of order $23$ are isomorphic to each other
2. Any two abelian groups of order $25$ are isomorphic to each other

Progress

I know that if $p$ is prime, up to isomorphism, there is only one group of order $p$, but I thought that only applies to small orders, less than $8$.

Answer 1

In virtue of the Cauchy's theorem for groups, every group with order $23$ is cyclic, hence isomorphic to $C_{23}$. On the other hand, $C_{25}$ and $C_5\times C_5$ are not isomorphic, since in the latter there is no element with order $25$.