# Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Suppose $G$ is a Finite abelian group (?) (i.e., it is both a finite group and an Abelian group (?)). Let $C(G)$ be the set of conjugacy classes of $G$ and $R(G)$ be the set of irreducible representations of $G$ over $\mathbb{C}$. The automorphism group $\operatorname{Aut}(G)$ acts on the sets $C(G)$ and $R(G)$. The claim is that the orbit sizes in both sets under the action of $\operatorname{Aut}(G)$ are the same. In other words, there is a size-preserving bijection between the sets of orbits in $C(G)$ and $R(G)$.