Let $G$ a finite group and $H,K$ subgroups of $G$. Show that if $[G:H\cap K]= [G:H][G:K]$ then $G=HK$.

I proved that $G=HK$ implies $[G:H\cap K]= [G:H][G:K]$ but not the other direction.

Thanks for your help.


Hint: Use the fact that $$|HK|=\frac{|H||K|}{|H\cap K|}.$$

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.