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Let be $G$ a group and $H$ and $K$ two subgroups such that $H\leq K \leq G$.

Let be $N\trianglelefteq G$. How can I prove that the relationions

$NH=KN\;$ and $\;H\cap N=K \cap N$ imply $H=K$?

Thanks for the help!

mesel
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GGG
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2 Answers2

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Apply Dedekind's Modular Law: if $H \leq K$, $NH=KN$ and $H \cap N = K \cap N$, then $$H=H(H \cap N)=H(K \cap N)= K \cap NH= K \cap KN=K.$$

Community
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Nicky Hekster
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  • What's happend with the hypothesis that $N\trianglelefteq G$? Don't was used? – Pitágoras Apr 09 '15 at 21:47
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    That is an excellent question! Yes it was used in a subtle way: the Dedekind Modular Law is about subgroups, not merely subsets. And $HN$ is a subgroup, because $N$ is normal. If $N$ wouldn't be normal, then $HN$ is not necessarily a subgroup. – Nicky Hekster Apr 10 '15 at 08:49
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Hint: $H/H \cap N\cong HN/N=KN/N\cong K/K\cap N$

mesel
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  • I know this is an old one, but still worth a shot. I tried doing it using the map $\varphi (h) = hN$ but I'm having trouble with showing that it's indeed a homomorphism. – Uria Mor Sep 26 '16 at 16:43