# What is Cosets: Definition and 68 Discussions

In mathematics, specifically group theory, a subgroup H of a group G may be used to decompose the underlying set of G into disjoint equal-size subsets called cosets. There are left cosets and right cosets. Cosets (both left and right) have the same number of elements (cardinality) as does H. Furthermore, H itself is both a left coset and a right coset. The number of left cosets of H in G is equal to the number of right cosets of H in G. This common value is called the index of H in G and is usually denoted by [G : H].
Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group G, the number of elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes.

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1. ### I What are double cosets in group theory?

Hi Pfs It is the first time that reas something about "double cosets" it was in this paper https://arxiv.org/pdf/0810.2091.pdf At page 4 i read ∆1\SU(3)/∆1 = ∆\U(3)/∆ Could you help to understand what are these sets (or cosets)? thanks
2. ### MHB Prove Lagrange’s Theorem for left cosets

Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...
3. ### Finding Cosets of subgroup <(3,2,1)> of G = S3

Homework Statement Find all cosets of the subgroup H in the group G given below. What is the index (G : H)? H = <(3,2,1)>, G = S3 Homework EquationsThe Attempt at a Solution I will leave out the initial (1,2,3) part of the permutation. We have S3 =...

31. ### Proof: Cosets equal or disjoint

Homework Statement Two left cosets aH, bH of H in G are equal if and only if a^{−1}b ∈ H. This is also equivalent to the statement b ∈ aH. Proof: Suppose that aH = bH. Then e ∈ H. So, b = be ∈ bH. If aH = bH then b ∈ aH. So, b = ah for some h ∈ H. But, solving for h, we get h = a −1 b...
32. ### Subgroup conjugation and cosets

Hello, I am having trouble with the following problem. Suppose that H is a subgroup of G such that whenever Ha≠Hb then aH≠bH. Prove that gHg^(-1) is a subset of H. I have tried to manipulate the following equation for some ideas H = Hgg^(-1) = gg^(-1)H but I don't know how to go...
33. ### Cosets and Vector Spaces Question

In studying vector spaces, I came across the coset of a vector space. We have an equivalence relation defined as u = v \rightarrow u-v \in W where W is a subspace of V. the equivalence class that u belongs to is u + W. I can see why u must belong to this equivalence class ( the...
34. ### MHB Lagrange thm: orbits as equivalence classes and cosets

Hi all, first post, please bear with me! I am trying to understand Lagrange's Theorem by working through some exercises relating to the Orbit-Stabilizer Theorem (which I also do not fully understand.) I think essentially I'm needing to learn how to show cosets are equivalent to other things or...
35. ### Help with proof regarding normal subgroups and cosets

Homework Statement I think I've got this one about figured out, I just wanted someone to check it over. (For this problem, (a-1) is a inverse, (b-1) b inverse, etc.) "Let G be a group, H a subgroup of G. Then, H is normal in G iff every left coset of H is equal to some right coset of H"...
36. ### Understanding Cosets and Subspaces in Linear Algebra

Hi, I have just begin with Linear Algebra. I came across cosets and I don't understand what is the difference between cosets and subspaces? thanx in advance.
37. ### I need a pretty basic explanation of cosets

So, I'm trying to self-teach myself Abstract Algebra, and this idea of cosets is killing me, and I'm not completely sure why. Basically, I think I understand the theory, but I'm having a hard time visualizing it. Does anyone know a basic example of a group that would have a different right...
38. ### Linar Codes and cacluating the cosets

Given a liear code generated by 01111, 11010 and 10100 how do you calculate the cosets of C? Does this mean because it is generated by that matrix that it is not my acutal code C so am I suppose to find C then find my cosets or do I straight up use the generated matrix as it is equivalent to C?
39. ### Exploring the Connection Between Cosets and Normal Subgroups

Hey guys I'm curious about how to interpret cosets and normal subgroups. I do know the definitions of both, but I do not understand how they relate to each other. A (left) coset is supposed to partition a group as well as normal subgroups, but I'm sure there is a more profound relationship...
40. ### Finding all of the right cosets of H in G

Homework Statement "Write out all the right cosets of H in G where G = (a) is a cyclic group of order 10 and H = (a^2) is the subgroup of G generated by a^2." Homework Equations - If G = (a), then G = {a^i | i=0,-1,1,-2,2...}. - A right coset is the set Hb = {hb | h is in H} - Order of...
41. ### What Are the Cosets in Q/Z(Q)?

Homework Statement Find the cosets in Q/Z(Q) Homework Equations The Attempt at a Solution So Z(Q) is the centre of Q.. Then Z(Q) is normal in Q. I don't get what the cosets would be without any given elements of Q or Z(Q).. But I'm assuming since it is the centre of Q there...
42. ### Group action on cosets of subgroups in non-abelian groups

This is not a homework question, just a general question. Let G be a non-abelian finite group, S < G a non-normal proper subgroup of index v >= 2, and G/S the set of v right cosets S_1 = S, S_2, ..., S_v, of S in G. We know there is a naturally defined right-multiplication action G x G/S -->...
43. ### Group Theory: Is A a Left Coset of G?

This is not a homework problem. I was just wondering. Let G be a group and let A be a finite subset of G. If |A²|=|A|² (where A^2=\{a_1a_2~\vert~a_1,a_2\in A\} ). Is it true that A is a left coset of G? If A has two elements, then I have proven that this is true. But for greater elements...
44. ### Proving something about right cosets of distinct subgroups of a group

Homework Statement Prove that a subset S of a group G cannot be a right coset of two different subgroups of G. Homework Equations The relevant equations are those involving the definitions of right cosets. a is in the right coset of subgroup H of group G if a = hg where h is in H and...
45. ### Abstract Algebra- A simple problem with Cosets

I need to find all the cosets of the subgroup H={ [0], [4], [8] ,[12] } in the group Z_16 and find the index of [Z16 : H]. Help would be appreciated :)
46. ### Finding Cosets of Subgroups in Groups

Homework Statement [PLAIN]http://img641.imageshack.us/img641/8448/63794724.gif The Attempt at a Solution Firstly, how do I list the elements of H? According to Lagrange's theorem if H is a subgroup of G then the number of distinct left (right) cosets of H in G is |G|\|H|. So I...
47. ### Proving Cosets of Subgroups in Nonabelian Finite Groups

Hi, This is not a homework question. I am a trying to prove a result for myself, and the question is can I always find, in a nonabelian finite group G, and some fixed proper subgroup S < G, two distinct elements, which we shall call x and y, outside of S, such that the cosets Sx = Sx^{-1}...
48. ### Theorem about cosets - what am I doing wrong

[b]1. The problem statement, all variables and given/known I am reading about cosets and am stuck on this proposition. Let H be a subgroup of a group G. If aH and bH have an element in common, then they are equal. But let the group be Z with addition as the law of composition. Let H be 5Z...
49. ### How many distinct H cosets are there?

Homework Statement Consider the cyclic group Cn = <g> of order n and let H=<gm> where m|n. How many distinct H cosets are there? Describe these cosets explicitly. Homework Equations Lagrange's Theorem: |G| = |H| x number of distinct H cosets The Attempt at a Solution |G| = n...
50. ### How do I find all cosets in GL(n) for the set H={A in GL(n) | det(A) = 1}?

Can someone explain to me how to find all the cosets of a set like H={A in GL(n) | det(A) = 1} in GL(n) (set of invertible n x n matrices)? It's obvious how to find all the cosets for something simple like 3Z (set of all multiples of 3) in Z, we just find elements in Z, but not in 3Z that...