Abstract Algebra: Index of Subgroups Here's the problem I'm working on:
Prove: Suppose $H$ has index $p$ and $K$ has index $q$, where $p$ and $q$ are distinct primes. Then the index of $H \cap K$ is a multiple of $pq$. (Plus: do you think this is still true when $G$ is not finite? Explain.)
We have been talking about the index of a subgroup and Lagrange's Theorem, but I am not quite getting it. Here's what I am thinking is the direction to go:
Using Lagrange's Theorem, we know that $|H \cap K| * [G:H \cap K] = |G|$. We also know this applies to both of the subgroups $H$ and $K$ such that $|H| * p = |G|$ and $|K| * q = |G|$. I need to find some way to prove that $[G:H \cap K]$ is equal to $p * q * x$ where $x$ is some integer such that $p * q * x = |G| / |H \cap K|$. 
And that is where I am at a loss. 
Thanks.

Suppose $A\le B\le C$. $\,$ Can you say anything about the indices $[C:A]$, $[C:B]$, $[B:A]$?
Hint: think of them as ratios of numbers, not groups, and make a claim about divisibility.
Now apply to your problem. Can you identify where two cases of $A\le B\le C$ occur implicitly in the problem setup? What can you conclude? Think arithmetic / elementary number theory.
