The square map is a group homomorphism precisely on abelian groups

Let G be a group. Show that the map \varphi : G \rightarrow G given by g \mapsto g^2 is a homomorphism if and only if G is abelian.


(\Leftarrow) Suppose G is abelian. Then \varphi(ab) = abab = a^2 b^2 = \varphi(a) \varphi(b), so that \varphi is a homomorphism.

(\Rightarrow) Suppose \varphi is a homomorphism. Then we have abab = \varphi(ab) = \varphi(a) \varphi(b) = aabb, so that $abab = aabb$. Left multiplying by a^{-1} and right multiplying by b^{-1}, we see that ab = ba. Thus G is abelian.

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