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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

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marked as duplicate by Seirios, Gerry Myerson, YACP, Alexander Gruber, Hagen von Eitzen Feb 9 '13 at 9:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

You can see this question:… – Seirios Feb 9 '13 at 8:42
Thank you Seirios, I have seen this question but is still not so clear to me. – Faye Feb 9 '13 at 8:58
up vote 2 down vote accepted

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?)

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for the first part also just working with the definitions work. This is just to clarify that there is nothing fancy needed for either parts. – Ittay Weiss Feb 9 '13 at 8:23
Agreed. I just have this penchant for trying to get students to think with homomorphisms. :-) – James Feb 9 '13 at 8:45
Thank you both for your feedback. – Faye Feb 9 '13 at 8:57
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
+1 for this and that maple code. Thanks James. – Babak S. Feb 9 '13 at 11:21

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