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

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite abelian group) must also satisfy the second group property (i.e., finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group)
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Statement

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).

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