Abelian implies every element is automorphic to its inverse

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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., abelian group) must also satisfy the second group property (i.e., group in which every element is automorphic to its inverse)
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Statement

Suppose G is an abelian group. Then, for any g \in G, there is an automorphism sending g to g^{-1}.

Facts used

  1. Inverse map is automorphism iff abelian

Proof

The proof follows from fact (1).