# Group as a product of subgroups

Let $G$ be a finite group, and $H,K$ be proper non-trivial subgroups such that $H\cap K=1$, and $HK=G$. Is it necessary that one of these subgroups is normal in $G$?

No, neither of the subgroups needs to be normal.

Take for example the alternating group $A_5$ which has a subgroup of order 12 (a copy of $A_4$) and one of order 5 (the 5-Sylow). Now clearly, these intersect trivially, as their orders are coprime. But since the product of their orders is exactly $60 = |A_5|$ this means that their product is all of $A_5$. Since $A_5$ is simple, neither of these can be normal.

• :-) $\quad +1\quad$ – Namaste Mar 21 '13 at 1:47