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

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?

marked as duplicate by Seirios, Gerry Myerson, user26857, Alexander Gruber♦, Hagen von EitzenFeb 9 '13 at 9:19

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?)
• Great, so its just using the definitions for the direct products of $n$ groups and not prove the axioms for a group to be abelian. Right. Cheers. – Faye Feb 9 '13 at 9:01