In direct products of $n$ groups, do we also prove conditions for a group to be abelian or not? [duplicate]

Question

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• Proving that Direct product $G_1 \times … \times G_n$ is abelian if and only if each of $G_1, …,G_n$ is abelian 2 answers

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

Answer 1

There are two parts that you need to prove, as follows (in either order).

First, assume that the direct product $G_{1}\times\cdots\times G_{n}$ is abelian, and then show that each of $G_{1},\ldots, G_{n}$ must also be abelian. (Hint: Find a homomorphism from the direct product onto an arbitrary $G_{i}$ and use that to help with the proof.)

Second, assume that all of $G_{1},\ldots, G_{n}$ are abelian, and show that the direct product $G_{1}\times\cdots\times G_{n}$ must be abelian. (Hint: Just use the definitions. What does a typical element of $G_{1}\times\cdots\times G_{n}$ look like?)