TITLE: What is an example of a situation where AB is not subgroup of G, when A, B are subgroups of G?
QUESTION [5 upvotes]: What is an example of a situation where AB is not subgroup of G, when A, B are subgroups of G?
My first instinct is always to go for some dihedral group or other...But I could not find an example of a case when the above is true. 
Maybe I am doing it wrong...

REPLY [2 votes]: This is an in page 40 from Hungerford's book Algebra, however; touchable examples above are more interesting to me.

Let $G$ be a group of order $p^{k}m,$ with $p$ prime and $(p,m)=1.$ Let $H$ be a subgroup of order $p^{k}$ and $K$ a subgroup of order $p^{d}$, with $0<d\leq k$ and $K\nsubseteq H.$ Show that $HK$ is not a subgroup of $G$.

You can solve it by using this fact that: $$\displaystyle |HK|=\frac{|H||K|}{|H\cap K|}$$