Let G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two statements is true or false. Give a proof or a counterexample in each case. (a) The group G × H is abelian if and only if G and H are both abelian. (b) The group G/N is abelian if and only if both G and N are abelian.

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A: Thanks for the question :)And your upvote will be really appreciable ;)

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A: As per the guidelines we have answered first three parts.For remaining parts please repost question.

Q: 3. Let G be a group with identity e. Show that if a ★a=e for all a in G, then G is Abelian.

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Q: (c) Prove that G/N is Abelian iff aba-¹b-¹ N a,b G. 2

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Q: 9. If : G→ H is a group homomorphism and G is abelian, prove that (G) is also abelian.

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