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. H

    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. AutGuy98

    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. Prof. 27

    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 =...
  4. karush

    MHB -412.7.3 decide if cosets of H are the same

    Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$. $\textit{ Use }$ $$aH=bH \textit{ or }aH\cap bH=\oslash$$ $\textit{then..}$ $$aH = bH \textit{ iff } (b-a) \textit{ is in } \textit{H}$$ to decide whether or not the following cosets of H are the same. $\textsf{a. 11 + H and 17 + H}$ $\textsf{b. -1 + H...
  5. karush

    MHB What are Left Cosets and How Do You Find Them?

    Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$ Find all the left cosets of $H \textit{ in } \Bbb{Z}$ok I can only see that From $\textit{H}$ we have $\textit{H}=3 \Bbb{Z}$ thus we cosets of $1+3\Bbb{Z},\quad 2+3\Bbb{Z} \cdots $ didn't know what the "left cosets" meantthere must be more that could be...
  6. Mr Davis 97

    I Enumerating the cosets of a kernel

    Suppose that we have, for purposes of example, the homomorphism ##\pi : \mathbb{R}^2 \to \mathbb{R}## such that ##\pi((x,y)) = x+y##. We see that ##\ker(\pi) = \{(x,y)\in \mathbb{R}^2 \mid x+y=0\}##. How can we enumerate all of the cosets of the kernel? My thought was that of course as we range...
  7. N

    Proving Normality of [0] in Z/3Z Quotient Group

    Homework Statement I am looking at the quotient group G = Z/3Z which is additive and abelian. The equivalence classes are: [0] = {...,0,3,6,...} [1] = {...,1,4,7,...} [2] = {...,2,5,8,...} I want to prove [0] is a normal subgroup, N, by showing gng-1 = n' ∈ N for g ∈ G and n ∈ N. Since G...
  8. F

    Elementary property of cosets

    Homework Statement Let G be a group, and H a subgroup of G. Let a and b denote elements of G. Prove the following: 1. ##Ha = Hb## iff ##ab^{-1} \epsilon H##. Homework Equations Let ##e_H## be the identity element of H. The Attempt at a Solution Proof: <= Suppose ##ab^{-1} \epsilon H##. Then...
  9. L

    A How many cosets are there when taking a subgroup in a group and forming cosets?

    When we take some subgroup ##H## in ##G##. And form cosets ##g_1H, g_2H,...,g_{n}H##. Is ##H## also coset ##eH##, where ##e## is neutral? So do we have here ##n## or ##n+1## cosets?
  10. Mr Davis 97

    I Group Theory: Multiplicative Cosets Explained

    In learning group theory, you learn about cosets as a partition of the entire group with respect to a subgroup. Since we are dealing with groups, there is only on operation that can be used to form the cosets. But when we come to rings, we now have two operations. So my question is, why do we...
  11. RJLiberator

    Write down the cosets (right/left) of this:

    Homework Statement Q: In Qu, write down the elements in all the right cosets and all the left cosets of <j>. Homework Equations Let H ≤ G. A coset of H is a subset of G of the form Hg for some g ∈ G. where ≤ denotes subgroup. The Attempt at a Solution I need help understanding what all...
  12. NoName3

    MHB Find Left Cosets of Subgroup in $\mathbb{Z}_{15}, D_4$

    How do I find the left cosets of: $(a)$ $\left\{ [0], [5], [10] \right\} \le \mathbb{Z}_{15}$ ($\mathbb{Z}_n$ is additive group modulo $n$). $(b)$ $\left\{e, y, y^2, y^3 \right\} \le D_4$ where $y$ denotes rotation of a square. The not equal to here denotes subgroup. The trouble I've with...
  13. S

    I Why only normal subgroup is used to obtain group quotient

    Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...
  14. mnb96

    Cosets of Monoids: Conditions for Partitions

    Hi, We know that given a group G and a subgroup H, the cosets of H in G partition the set G. Now, if instead of groups we consider a monoid M and a submonoid H, the cosets of H in M in general do not partition the set M. However, are there some conditions that we can impose on H under which...
  15. M

    Question about Cosets: Does abH=baH imply ab=ba? | Group Theory Homework

    Homework Statement I really just need clarification about a property of cosets. I can't find anything explicitly stating one way or the other, and it could be because I'm wrong, or because it's deemed trivially true. Homework Equations Left Coset: (aH)(bH)=abH, where a,b are elements of a...
  16. PsychonautQQ

    Question about Cosets and Lagrange's Theorem

    If a is an element of G and H is a subgroup of H, let Ha be the right coset of H generated by a. is Ha a subgroup? I have this question because i feel like the answer should be know, yet my textbook notation makes it look like it is. Why I think it should only be a set: Let G = <a> and H =...
  17. S

    Counting Cosets in Abstract Algebra | Pinter's Self Study

    Hi, I am doing self study of Abstract Algebra from Pinter. My doubt is regarding Chap 13 Counting Cosets: A coset contains all products of the form "ah" where a belongs to G and h belongs to H where H is a subgroup of G. So each coset should contain the number of elements in H. Now the number of...
  18. B

    Proof of Left Cosets Partition a Group

    Here is a link to a proof which I am trying to understand. http://groupprops.subwiki.org/wiki/Left_cosets_partition_a_group The claim I am referring to is number 4, which is Any two left cosets of a subgroup either do not intersect, or are equal. Assuming that I am skeptical, then for all I...
  19. PsychonautQQ

    Modern Algebra: Basic problem dealing with Cosets

    Homework Statement If H is a subgroup of G and Ha = bH for elements a and b in G, show that aH = Hb. Homework Equations None needed The Attempt at a Solution I've basically just been fiddling around by right and left side multiplication of inverses and what not and can't seem to get it...
  20. O

    MHB Counting Cosets: Clarifying Right & Left Cosets

    i am reading a chapter on counting cosets and I am not sure i fully understand the theory behind right and left cosets. can i please be given clear descriptions perhaps with examples.
  21. Z

    Cosets: difference between these two statements

    Hi all, Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G. What is the difference between these two statements: 1) for a given aH in G/H, the set {g in G : fg(aH) = aH } 2) set {g in G : fg = the...
  22. Z

    Abstract Algebra: Solving with Cosets

    Homework Statement Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G. I know that fg is a well-defined permutation. However, we have not established (yet) that G/H is a group. 2 parts to the...
  23. C

    Understanding a proof about groups and cosets

    Homework Statement H is a subgroup of G, and a and b are elements of G. Show that Ha=Hb iff ab^{-1} \in H . The Attempt at a Solution line 1: Then a=1a=hb for some h in H. then we multiply both sides by b inverse. and we get ab^{-1}=h This is a proof in my book. My question is...
  24. P

    Relationship between orbits and cosets

    How are orbits and cosets related? Are all orbits cosets? Are all cosets orbits? Also, what exactly are G-sets and G-equivariant sets?
  25. U

    Proving Existence of Integer n for Left Cosets in Q/Z | Homework Solution

    Homework Statement View Z as a subgroup of the additive group of rational numbers Q. Show that given an element \bar{x} \in Q/Z there exists an integer n \geq 1 such that n \bar{x} = 0. Homework Equations The Attempt at a Solution As we are working in an additive group, it is...
  26. J

    The set of all conjugates of a and the set of all cosets of the Centralizer of a

    I am working on constructing a bijection between the set of all conjugates of a and the set of all cosets of the Centralizer of a. Now, I let [a]={xεG: xax-1}. This is the set of all conjugates of a. The set {Cax : xεG} is the set of all cosets of Ca. Hence, I want a function f: [a] -> {Cax : x...
  27. E

    Exploring the Additive Cosets of an Ideal in R

    Let R be the subring {x + yi : x, y in 2Z} of C, and let I be the ideal {x + yi : x,y in 2Z}of R. How many additive cosets has I in R? List them clearly. I know definition of ideal but ı don't know how to write in set is that question describe.Please help :)
  28. S

    Exploring Cosets of R=Z_4[x]/((x^2+1)*Z_4[x])

    I'm trying to list the cosets of the following ring and describe the relations that hold between these cosets. R=Z_4[x]/((x^2+1)*Z_4[x]) I'm using the division algorithm since x^2+1 is monic in the ring Z_4[x].Now for every f that belongs to Z_4[x] by the division algorithm...
  29. S

    MHB Exploring Cosets of a Ring with Division Algorithm

    I'm trying to list the cosets of the following ring and describe the relations that hold between these cosets. R=Z_4[x]/((x^2+1)*Z_4[x]) I'm using the division algorithm since x^2+1 is monic in the ring Z_4[x].Now for every f that belongs to Z_4[x] by the division algorithm...
  30. S

    MHB "structure" on the cosets → normal?

    Let $H\leq G$, where $G$ is some infinite group, and there exists some $g\in G$ such that the set $\{g^n: n\in\mathbb{Z}\}$ is a transversal for $G/H$. Then is $H$ normal in $G$? I suspect not. However, I cannot seem to find a counter-example. (By "a transversal for $G/H$" I mean that 1)...
  31. L

    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. T

    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. M

    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. B

    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. S

    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. A

    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. A

    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. R

    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. W

    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. H

    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. M

    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. N

    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. micromass

    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. A

    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. G

    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. R

    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. N

    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. M

    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. K

    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. L

    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...
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