Let $H$ and $K$ be subgroups of a group $G$. Suppose $H=A\times B$.
Does it follow that $H\cap K=(A\cap K)\times (B\cap K)$?
I'm having a hard time trying to prove that $H\cap K\le (A\cap K).(B\cap K)$.
EDIT: The question stated in this form has negative answer. I should have added the assumption that $A\cap K$ in non-trivial!