What is Quotient groups: Definition and 30 Discussions
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.
In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N", where "mod" is short for modulo.)
Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects. For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.
Hello,
I have a question that I would like to ask here.
Let ##L = \left\{ x \in \mathbb{Z}^m : Ax = 0 \text{ mod } p \right\}##, where ##A \in \mathbb{Z}_p^{n \times m}##, ##rank(A) = n##, ## m \geq n## and ##Ax = 0## has ##p^{m-n}## solutions, why is then ##|L/p\mathbb{Z}^m| = p^{m-n}##?
I...
I want to identify:
##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##.
##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##.
The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer.
In 1 how to define the...
Of course it must be an infinite group, otherwise |G/N|=|G|/|N| and then {e} is the only ( and trivial) solution. I understand there is a result that for every quotient Q:=G/N there is a subgroup H that is isomorphic to Q. Is that the case?
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...
So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following:
Consider the Klein four group, if we...
Hello,
I am having some trouble truly interpreting what certain notation means when defining quotient groups, etc. (My deepest apologies in advance, with my college workload I simply have not had the time to really sit down and master latex.) Here are a few random examples I've seen in...
I have just received some help from Euge regarding the proof of part of the Correspondence Theorem (Lattice Isomorphism Theorem) for groups ...
But Euge has made me realize that I do not understand quotient groups well enough ... here is the issue coming from Euge's post ...
We are to consider...
Assume that G is some group with two normal subgroups H_1 and H_2. Assuming that the group is additive, we also assume that H_1\cap H_2=\{0\}, H_1=G/H_2 and H_2=G/H_1 hold. The question is that is G=H_1\times H_2 the only possibility (up to an isomorphism) now?
G is a group and H is a normal subgroup of G.
where G=Z6 and H=(0,3)
i was told to list the elements of G/H
I had:
H= H+0={0,3}
H+1={14}
H+2={2,5}
now they are saying H+3 is the same as H+0, how so?
In Beachy and Blair: Abstract Algebra, Section 3.8 Cosets, Normal Groups and Factor Groups, Exercise 17 reads as follows:
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17. Compute the factor group ( \mathbb{Z}_6 \times...
I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms.
Exercise 17 in Section 3.1 (page 87) reads as follows:
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Let G be the dihedral group od order 16.
G = <...
The definition given is...
"Let ##\phi: G \rightarrow H## be a homomorphism with kernel ##K##. The quotient group ##G/K## is the group whose elements are the fibers (sets of elements projecting to single elements of H) with group operation defined above: namely if ##X## is the fiber above...
Homework Statement
Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there?
The Attempt at a Solution
I don't even know where to start with this question. If someone could just point me in the right direction that would be...
Homework Statement
I have attached the problem below.
Homework Equations
The Attempt at a Solution
I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...
Homework Statement
Show that Z18/M isomorphic to Z6 where m is the cyclic subgroup <6>
operation is addition
The Attempt at a Solution
M = <6> , so M = {6, 12, 0}
I figured I could show that Z18/M has 6 distinct right cosets if I wanted to do M + 0 = {6, 12, 0} M + 1 = {7, 13, 1}...
hey guys,
I just want grasp the whole concept of quotient groups,
I understand say, D8/K where K={1,a2}
I can see the quotient group pretty clearly without much trouble however I start to get stuck when working with larger groups, say S4
For instance S4/L where L is the...
Hi I am trying to learn about quotient groups to fill the gaps on things I didn't quite understand from undergrad. Anyway I have a question regarding this:
Can someone please explain how { 0, 2 }+{ 1, 3 }={ 1, 3 } in Z4/{ 0, 2 }?
I would think since 0 + 1 = 1 and 2 + 3 = 1 under mod 4...
Homework Statement
Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) }
and N=<(13)(24)> which is a normal subgroup of d4 .
List the elements of d4/N .
Homework Equations
The Attempt at a Solution
I computed the left and right cosets to...
Homework Statement
Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite
Homework Equations
|G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G.
The Attempt at a Solution
I...
Homework Statement
Let G be a finite group and N\triangleleftG such that |N| = n, and gcd(n,[G:N]) = 1.
Proof that if x^{n} = e then x\inN.
Homework Equations
none.
The Attempt at a Solution
I defined |G| = m and and tried to find an integer which divides both n and m/n.
I went for some X...
As a way to keep busy in between semesters I decided to work my way through Algebra by Dummit and Foote in order to prepare for the fall. Working my way through quotient groups is proving to be quite difficult and as a result I'm stuck on an exercise that looks simple, but I just don't know...
Homework Statement
Let G be the group {
\begin{bmatrix}{a}&{b}\\{0}&{c}\end{bmatrix}
| a, b, c are in Z_p with p a prime}
Then let K = {
\begin{bmatrix}{1}&{b}\\{0}&{1}\end{bmatrix}
| b in Z_p}
The map P: G --> Z*p x Z*p is defined by
P(...
Homework Statement
Show that every element of the quotient group \mathbb{Q}/\mathbb{Z} has finite order but that only the identity element of \mathbb{R}/\mathbb{Q} has finite order.
The Attempt at a Solution
The first part of the question I solved. Since each element of...
Homework Statement
Write the multiplication table of C_{6}/C_{3}
and identify it as a familiar group.
Homework Equations
The Attempt at a Solution
C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5}
C3={1,\omega,\omega^2}
The cosets are C3 and \omega^3C3
I just need help...
Find, up to isomorphism, all possible quotient groups of D6 and D9, the dihedral group of 12 and 18 elements.
First of all, I don't understand the question by what they mean about "up to isomorphism." Does this mean by using the First Isomorphism Theorem? Also does this question imply that...
Homework Statement
In Z4 x Z4, find two subgroups H and K of order 4 such that H is not isomorphic to K, but (Z4 x Z4)/H isomorphic (Z4 x Z4)/K
Homework Equations
The Attempt at a Solution
I know (Z4 x Z4) has twelve elements (0,0), (1,0), (2,0), (3,0), etc. I can generate subgroups of...
1. Show that every element of the quotient group G = Q/Z has finite
order. Does G have finite order?
he problem statement, all variables and given/known data
[b]2. This is the proof
The cosets that make up Q/Z have the form Z + q,
where q belongs to Q. For example, there is a...
Let G be a group and let N\trianglelefteq G , M\trianglelefteq G be such that N \le M. I would like to know if, in general, we can identify G/M with a subgroup of G/N.
Of course the obvious way to proceed is to look for a homomorphism from G to G/N whose kernel is M, but I can't think of...
Homework Statement
Let H and K be normal subgroups of a group G. Give an example showing that we may have H isomorphic to K while G/H is not isomorphic to G/K.
Homework Equations
The Attempt at a Solution
I don't want to look in the back of my book just yet. Can someone give me a...