$[G : H] < ∞$ and$[G : K] < ∞$ then $[G : H ∩ K] < ∞$ where $H$ and $K$ be subgroups of $G$ [duplicate]

Let $H$ and $K$ be subgroups of a group $G$. Then is the following true ?

If $[G : H] < ∞$ and $[G : K] < ∞$, then $[G : H ∩ K] < ∞$.

I think this's false because there's still a case that $H ∩ K$ could be empty. But the textbook requires to prove this statement. So I am a bit confusing on this point.. any help?

• The intersection of two subgroups can't be empty. – Gerry Myerson Apr 19 '18 at 7:31
• @GerryMyerson Because of $e$? – delinco Apr 19 '18 at 7:33
• – Gerry Myerson Apr 19 '18 at 7:34
• @GerryMyerson better be marked as duplicate – delinco Apr 19 '18 at 7:39