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Suppose a group $G$ with $|G|=162$ has a normal subgroup $K$ with $|K|=9$. Show that $G$ must have a subgroup of order 18, using $HK/K \cong H/(K \cap H)$.

I feel like this should be very easy, and my ongoing failure to do it is beginning to make me really resent mathematics.

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Notice first that $G$ has some subgroup $H$ of order $2$. Then show that $HK$ has order $18$.

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Ahh, yes, thank you so very much---I have an exam tomorrow and I'm freaking out a little bit. – Switch Nov 15 '11 at 13:57

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