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.

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  1. A

    I What is the size of the quotient group L/pZ^m?

    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...
  2. A

    I Using the orbit-stabilizer theorem to identify groups

    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...
  3. W

    I Can a group be isomorphic to one of its quotients?

    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?
  4. 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...
  5. W

    I Understanding Quotient Groups in Abstract Algebra

    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...
  6. M

    MHB Quotient Groups & how to interpret notation?

    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...
  7. Math Amateur

    MHB Arithmetic for Quotient Groups - How exaclty does it work

    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...
  8. J

    Two quotient groups implying Cartesian product?

    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?
  9. O

    MHB Understanding Finite Quotient Groups: G/H with G=Z6 and H=(0,3)

    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?
  10. Math Amateur

    MHB Direct Products and Quotient Groups

    In Beachy and Blair: Abstract Algebra, Section 3.8 Cosets, Normal Groups and Factor Groups, Exercise 17 reads as follows: ---------------------------------------------------------------------------------------------------------------------- 17. Compute the factor group ( \mathbb{Z}_6 \times...
  11. Math Amateur

    MHB Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

    I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms. Exercise 17 in Section 3.1 (page 87) reads as follows: ------------------------------------------------------------------------------------------------------------- Let G be the dihedral group od order 16. G = <...
  12. R

    Help Understanding Quotient Groups? (Dummit and Foote)

    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...
  13. P

    Finding subgroups of Factor/ Quotient Groups

    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...
  14. T

    Product of Quotient Groups Isomorphism

    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...
  15. T

    Show Quotient Groups are isomorphic

    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}...
  16. J

    Quotient groups of permutations

    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...
  17. D

    Question about wiki artical on Quotient Groups

    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...
  18. F

    Confused about Quotient groups

    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...
  19. L

    Orders of Quotient Groups (Abstract Algebra)

    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...
  20. Z

    Quotient groups related problem

    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...
  21. P

    Quotient Groups and their Index

    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...
  22. K

    Normal subgroups, quotient groups

    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(...
  23. 3

    Finite Order in Quotient Groups: Q/Z and R/Q

    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...
  24. G

    Mutliplication table of quotient groups

    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...
  25. Z

    Exploring Quotient Groups of D6 & D9

    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...
  26. D

    Isomorphic Quotient Groups in Z4 x Z4

    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...
  27. B

    Proof of Finite Order of G in Quotient Group Q/Z

    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...
  28. H

    Can We Identify Quotient Groups as Subgroups of the Original Group?

    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...
  29. E

    Isomorphic Quotient Groups: A Counterexample

    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...
  30. M

    Define R/Q: How to Add in Quotient Groups

    how to define R\Q?(under addition) R\Q={a+Q:? <a<?} a€R but if it is not bounded then it will repeat please help me n