# Index of Intersection of Subgroups with Finite Index

• Bashyboy
In summary, for subgroups of finite index in a possibly infinite group, the theorem of Lagrange can be used to show that the lower bound for |G : H ∩ K| is lcm(m,n). However, this theorem only applies to finite groups.
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##.

## 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!

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|##.

fresh_42 said:
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.

Bashyboy said:
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.

## 1. What is an index of intersection of subgroups with finite index?

The index of intersection of subgroups with finite index refers to the number of cosets in the intersection of two subgroups, divided by the product of the number of cosets in each subgroup. It is a measure of how much the two subgroups overlap within the larger group.

## 2. How is the index of intersection of subgroups with finite index calculated?

The index of intersection can be calculated using the formula [G : (H ∩ K)] = [G : H] * [G : K], where G is the larger group, H and K are the two subgroups, and [G : H] denotes the index of subgroup H in G.

## 3. What does a finite index of intersection imply?

A finite index of intersection indicates that the two subgroups have a finite number of cosets in their intersection. This can provide insights into the structure of the larger group and its subgroups.

## 4. Is the index of intersection of subgroups with finite index always a finite number?

Not necessarily. In some cases, the index of intersection may be infinite, indicating that the two subgroups have an infinite number of cosets in their intersection. This can occur if the two subgroups are not discrete within the larger group.

## 5. How is the index of intersection of subgroups with finite index related to group homomorphisms?

The index of intersection of subgroups with finite index plays a key role in the theory of group homomorphisms. It can be used to determine the size and structure of the image and kernel of a group homomorphism, and to prove important theorems such as the first isomorphism theorem.

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