# Index of Intersection of Subgroups with Finite Index

1. Jan 24, 2017

### Bashyboy

1. The problem statement, all variables and given/known data
Suppose that $H$ and $K$ are subgroups of finite index in the (possibly infinite) group $G$ with $|G : H|m$ and $|G:K|=n$. Prove that $lcm(m,n) \le |G : H \cap K | < mn$.

2. Relevant equations

3. The attempt at a solution

I was able to get the upper bound on $|G : H \cap K|$, but am having difficulty showing that the lower bound is $lcm(m,n)$. I tried showing that $m$ and $n$ both divide $|G : H \cap K |$, but I couldn't get anywhere. I could use some hints!

2. Jan 26, 2017

### Staff: Mentor

I think you can use the theorem of Lagrange to get $|G : H \cap K|=|G : K| \cdot |K : H \cap K|$ so $n\,\vert \,|G : H \cap K|$ and similar $m\,\vert \,|G : H \cap K|$.

3. Jan 27, 2017

### Bashyboy

Unfortunately, Lagrange's theorem only applies to finite groups.

4. Jan 27, 2017

### Staff: Mentor

I haven't checked the proof, but on the Wiki page it has been first stated ($|G|=|G : H| \cdot |H|$) and then appended "Especially for $|G|<\infty \; \ldots$" so I assumed that finiteness of $G$ isn't really required.