Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about this.
(1) [Converse] If I know that a finite group of nonprime order $n$ is always cyclic, can we conclude that $n$ is a prime or 1? In other words, if $n$ is not a prime or 1, can we always find a group of order $n$ which is not cyclic?
(2) If I know that a finite group of nonprime order $n$ is always unique up to isomorphism, can we conclude that every finite group of order $n$ is cyclic as well?