Are there finite groups $G$ and $H$ such that:
- $G$ is abelian.
- $H$ is nonabelian.
- for every $d\mid n$, $G$ and $H$ have the same number of elements of order $d$. ?
I know two finite abelian groups with the same numbers of elements of each order; are isomorphic. And there are nonabelian groups with the same property which are not isomorphic.