An automorphism of a group of odd order

#group_theory

Let $G$ be a finite abelian group of odd order. Let $ϕ : G \to G$ be the function defined by $ϕ (g) = g^{2}$ for all $g \in G$ . Prove that $ϕ$ is an automorphism.

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Let $G$ be a finite abelian group of odd order. Let $\phi:G\to G$ be the function defined by $\phi(g)=g^2$ for all $g\in G$. Prove that $\phi$ is an automorphism.