Index of Intersection of Subgroups with Finite Index

[Bashyboy]

Homework Statement

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 $lcm(m,n) \le |G : H \cap K | < mn$.

Homework Equations

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!

 

[fresh_42]

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 $m\,\vert \,|G : H \cap K|$.

 

[Bashyboy]

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 $m\,\vert \,|G : H \cap K|$.

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

 

[fresh_42]

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

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.