COSETS are equal for finite groups

In summary: It's a bit more involved than just proving that there are as many left cosets as right cosets, but it's a very effective way to solve it. That's a great way to solve to problem.
  • #1
mathwhiz22
4
0

Homework Statement



Prove that if H is a subgroup of a finite group G, then the number of right cosets of H in G equals the number of left cosets of H in G


Homework Equations



Lagrange's theorem: for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G.

The Attempt at a Solution



I know how to find subgroups of groups, and how to get the cosets from there. But i just don't understand how to show that right and left cosets will be equal because the group is finite..

Im stuck :( thanks for any help!
 
Physics news on Phys.org
  • #2
Try reading the proof of Lagrange's theorem.
 
  • #3
well, i know that since cosets from the subgroup of H form partitions of the group G, and G is finite, then G is completely separated into a finite number of cosets. I know that since each coset has |H| elements, then |G|=|H|*n so therefore |H| divides |G|, proving the theorem. SO, what does this have to say about left and right cosets being equal?
 
  • #4
What is n?
 
  • #5
n is the number of cosets of G
 
  • #6
And then there was light :).
 
  • #7
i still don't get it.. haha sorry :(
 
  • #8
mathwhiz22 said:
well, i know that since cosets from the subgroup of H form partitions of the group G, and G is finite, then G is completely separated into a finite number of cosets. I know that since each coset has |H| elements, then |G|=|H|*n so therefore |H| divides |G|, proving the theorem. SO, what does this have to say about left and right cosets being equal?

Good. Do the right cosets have the same properties? Think about how you might obtain a right coset given a left coset.
 
  • #9
Technically you should re-word the proof to specify which cosets you're working with, i.e. left or right. So in fact n = number of left cosets (if you used left cosets), or n = number of right cosets (if you used right cosets).

Evidently it doesn't matter, so there must be as many left as right cosets.
 
  • #10
morphism said:
Evidently it doesn't matter, so there must be as many left as right cosets.

If there are as many left and right cosets, then it does not matter. However, your remark is begging the question.
 
  • #11
Have you already proven, or can you prove, that all left cosets have the same number of members and that all right cosets have the same number of members? That's pretty straight forward. Have you proved that both left cosets and right cosets "partition G" (every member of G is in exactly one coset). It's obvious that if x is a member of H, xH and Hx are just equal to order of H. If every left coset has |H| members and G has |G| members, and left cosets partition G, how many left cosets are there? Same thing for right cosets.
 
  • #12
I am working on the same problem for my modern algebra class, and I think I have the right answer. However, one of my partners in my group has gone a different way in proving this which I'm not sure if it correct or not. She goes through Lagrange's theorem exactly, and then defines a mapping, alpha, from H to Ha and shows that it is onto and one-to-one.

Is that a proper way to prove this problem?
 
  • #13
swartzism said:
I am working on the same problem for my modern algebra class, and I think I have the right answer. However, one of my partners in my group has gone a different way in proving this which I'm not sure if it correct or not. She goes through Lagrange's theorem exactly, and then defines a mapping, alpha, from H to Ha and shows that it is onto and one-to-one.

Is that a proper way to prove this problem?

That's a great way to solve to problem.
 

What are cosets and why are they important in finite groups?

Cosets are subsets of a group that contain all elements that are obtained by multiplying a fixed element of the group by elements of a subgroup. They are important because they help us understand the structure and symmetry of finite groups.

How do you determine if two cosets are equal for finite groups?

Two cosets are equal if they contain the same elements. This can be determined by multiplying each element in one coset by the same element in the other coset, and if the resulting elements are all in both cosets, then they are equal.

Can two different subgroups have the same cosets in a finite group?

Yes, it is possible for two different subgroups to have the same cosets in a finite group. This is because cosets are defined by multiplying a fixed element by elements of a subgroup, and different subgroups can have the same elements when multiplied by the same fixed element.

What is the significance of having equal cosets in a finite group?

Having equal cosets in a finite group means that there is a strong relationship between the two subgroups. This can help us understand the structure of the group and can provide insights into its symmetry and properties.

How do cosets relate to Lagrange's Theorem in finite groups?

Cosets are important in Lagrange's Theorem, which states that the order of a subgroup must divide the order of the group. Cosets help us prove this theorem by showing that the number of distinct cosets is equal to the index of the subgroup, which must divide the order of the group.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
21
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
23
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
Back
Top