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If we let $G_1,...,G_n$ be groups,
When proving that the direct product $G_1 \times .... \times G_n$ is abelian if and only if each of $G_1,...,G_n$ is abelian, can someone please help me Im concerned about whether it should also prove that it holds for the conditions of a group to be abelian(inverse, unit element ...) or just prove straightforward that left hand side is true iff right hand side is?
Can someone please help clarify this. Thanks