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Does anybody know how I would go about proving this question ?

Let G and H be groups and let φ : G −→ H be a group homomorphism. Suppose that G is abelian and φ is a surjection. Prove that H is abelian.

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closed as off-topic by Arnaud D., Thomas Shelby, user1729, verret, Alexander Gruber Feb 25 at 7:18

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    $\begingroup$ Try writing out what it means to be surjective, and write out what you want to show. $\endgroup$ – Dave Feb 21 at 16:46
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    $\begingroup$ You need to provide more context or other details, and your question (whether interesting or not!) is likely to be closed if you do not edit your question to provide additional context. This context should ideally explain why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc. $\endgroup$ – user1729 Feb 21 at 17:05
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$$\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a)$$

Question 1: Where did I use the fact that $\phi$ is a homomorphism?

Question 2: Where did I use the fact that $G$ is abelian?

Question 3: How can you use surjectivity of $\phi$ to show that the above is enough to prove that $H$ is abelian?

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Another approach: As $G$ is abelian, $G/\ker(\varphi)$ is also abelian. But $H \cong G/\ker(\varphi)$ by the isomorphism theorem.

This also conveys why we are interested in maps - they preserve properties.

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  • $\begingroup$ I like your answer more than mine! Very clever. $\endgroup$ – Pawel Feb 22 at 17:00