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Let $G$ be a group, $H$ a subgroup of $G$, and $N$ a normal subgroup of $G$.

Verify that $HN=\{hn\mid h \in H, n \in N\}$ is a subgroup of $G$.

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    Hi, what have you tried so far?2012-05-21
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    I thought I might remind you: in order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many users find the use of the imperative ("Prove", "Show", etc) to be rude when asking for help. Please consider rewriting your post.2012-05-21
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    I know I need to show that HN satisfy the three axioms. 1.closed under the operation, identity and inverses. I am stuck in the first one.2012-05-21
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    Are you familiar with the result that says that if $H$ and $K$ are subgroups of $G$, then $HK$ is a subgroup if and only if $HK=KH$ as sets?2012-05-21
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    yes, I proved it few days ago2012-05-21
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    @anilorap: Then why not use that, instead of trying to work from scratch? Is it true that $$HN=\{hn\mid h\in H,n\in N\}\stackrel{?}{=}\{n'h'\mid n'\in N, h'\in H\} = NH\ ?$$ Also: your title had nothing to do with your question!2012-05-21
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    Sorry You are Right. I am doing Homework and I am missing 2 out of 10.. I wrote the little of the other exercise.2012-05-21
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    Got it. wow... Great..2012-05-21

1 Answers 1

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Hint. For subgroups $H$ and $K$ of $G$, $HK$ is a subgroup of $G$ if and only if $HK=KH$ as sets.

What happens when one of the two subgroups is normal?

Hint the alternative. For all $a,b,c\in G$, $abc = b(b^{-1}ab)c$.