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.
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
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...
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 =...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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 =...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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 :)
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...
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...
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)...
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...
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...
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...
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...
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"...
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.
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...
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?
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...
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...
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...
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 -->...
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...
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...
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 :)
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...
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}...
[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...
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...
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...