Algebraic structure → Group theoryGroup theory |
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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.

- Definition and illustration
- Definition
- Example: Addition modulo 6
- Motivation for the name "quotient"
- Examples
- Even and odd integers
- Remainders of integer division
- Complex integer roots of 1
- The real numbers modulo the integers
- Matrices of real numbers
- Integer modular arithmetic
- Integer multiplication
- Properties
- Quotients of Lie groups
- See also
- Notes
- References

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.

Given a group *G* and a subgroup *H*, and an element *a* ∈ *G*, one can consider the corresponding left coset: *aH* := { *ah* : *h* ∈ *H* }. Cosets are a natural class of subsets of a group; for example consider the abelian group *G* of integers, with operation defined by the usual addition, and the subgroup *H* of even integers. Then there are exactly two cosets: 0 + *H*, which are the even integers, and 1 + *H*, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).

For a general subgroup *H*, it is desirable to define a compatible group operation on the set of all possible cosets, { *aH* : *a* ∈ *G* }. This is possible exactly when *H* is a normal subgroup, see below. A subgroup *N* of a group *G* is normal if and only if the coset equality *aN* = *Na* holds for all *a* ∈ *G*. A normal subgroup of *G* is denoted *N* ◁ *G*.

Let *N* be a normal subgroup of a group *G*. Define the set *G*/*N* to be the set of all left cosets of *N* in *G*. That is, *G*/*N* = {*aN* : *a* ∈ *G*}. Since the identity element *e* ∈ *N*, *a* ∈ *aN*. Define a binary operation on the set of cosets, *G*/*N*, as follows. For each *aN* and *bN* in *G*/*N*, the product of *aN* and *bN*, (*aN*)(*bN*), is (*ab*)*N*. This works only because (*ab*)*N* does not depend on the choice of the representatives, *a* and *b*, of each left coset, *aN* and *bN*. To prove this, suppose *xN* = *aN* and *yN* = *bN* for some *x*, *y*, *a*, *b* ∈ *G*. Then

- (
*ab*)*N*=*a*(*bN*) =*a*(*yN*) =*a*(*Ny*) = (*aN*)*y*= (*xN*)*y*=*x*(*Ny*) =*x*(*yN*) = (*xy*)*N.*

This depends on the fact that *N* is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on *G*/*N*.

To show that it is necessary, consider that for a subgroup *N* of *G*, we have been given that the operation is well defined. That is, for all *xN* = *aN* and *yN* = *bN,* for *x*, *y*, *a*, *b* ∈ *G*, (*ab*)*N* = (*xy*)*N.*

Let *n* ∈ *N* and *g* ∈ *G*. Since *eN* = *nN,* we have, *gN* = (*eg*)*N* = (*ng*)*N.*

Now, *gN* = (*ng*)*N* ⇔ *N* = *g*^{−1}(*ng*)*N* ⇔ *g*^{−1}*ng* ∈ *N* ∀ *n* ∈ *N* and *g* ∈ *G*.

Hence *N* is a normal subgroup of *G*.

It can also be checked that this operation on *G*/*N* is always associative. *G*/*N* has identity element *N* and the inverse of element *aN* can always be represented by *a*^{−1}*N*. Therefore, the set *G*/*N* together with the operation defined by (*aN*)(*bN*) = (*ab*)*N* forms a group, the quotient group of *G* by *N*.

Due to the normality of *N*, the left cosets and right cosets of *N* in *G* are the same, and so, *G*/*N* could have been defined to be the set of right cosets of *N* in *G*.

For example, consider the group with addition modulo 6: *G* = {0, 1, 2, 3, 4, 5}. Consider the subgroup *N* = {0, 3}, which is normal because *G* is abelian. Then the set of (left) cosets is of size three:

*G*/*N*= {*a*+*N*:*a*∈*G*} = { {0, 3}, {1, 4}, {2, 5} } = { 0+*N*, 1+*N*, 2+*N*}.

The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.

The reason *G*/*N* is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.

To elaborate, when looking at *G*/*N* with *N* a normal subgroup of *G*, the group structure is used to form a natural "regrouping". These are the cosets of *N* in *G*. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.

Consider the group of integers **Z** (under addition) and the subgroup 2**Z** consisting of all even integers. This is a normal subgroup, because **Z** is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group **Z**/2**Z** is the cyclic group with two elements. This quotient group is isomorphic with the set {0,1} with addition modulo 2; informally, it is sometimes said that **Z**/2**Z***equals* the set {0,1} with addition modulo 2.

**Example further explained...**

- Let remainders of when dividing by .
- Then when is even and when is odd.
- By definition of , the kernel of ,
- ker() , is the set of all even integers.
- Let ker().
- Then is a subgroup, because the identity in , which is , is in ,
- the sum of two even integers is even and hence if and are in , is in (closure)
- and if is even, is also even and so contains its inverses.
- Define / H as for
- and / H is the quotient group of left cosets; / H.
- By the way we have defined , is if is odd and if is even.
- Thus, is an isomorphism from / H to .

A slight generalization of the last example. Once again consider the group of integers **Z** under addition. Let *n* be any positive integer. We will consider the subgroup *n***Z** of **Z** consisting of all multiples of *n*. Once again *n***Z** is normal in **Z** because **Z** is abelian. The cosets are the collection {*n***Z**, 1+*n***Z**, ..., (*n*−2)+*n***Z**, (*n*−1)+*n***Z**}. An integer *k* belongs to the coset *r*+*n***Z**, where *r* is the remainder when dividing *k* by *n*. The quotient **Z**/*n***Z** can be thought of as the group of "remainders" modulo *n*. This is a cyclic group of order *n*.

The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group *G*, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup *N* made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group *G*/*N* is the group of three colors, which turns out to be the cyclic group with three elements.

Consider the group of real numbers **R** under addition, and the subgroup **Z** of integers. Each coset of **Z** in **R** is a set of the form *a*+**Z**, where *a* is a real number. Since *a _{1}*+

If *G* is the group of invertible 3 × 3 real matrices, and *N* is the subgroup of 3 × 3 real matrices with determinant 1, then *N* is normal in *G* (since it is the kernel of the determinant homomorphism). The cosets of *N* are the sets of matrices with a given determinant, and hence *G*/*N* is isomorphic to the multiplicative group of non-zero real numbers. The group *N* is known as the special linear group SL(3).

Consider the abelian group **Z**_{4} = **Z**/4**Z** (that is, the set { 0, 1, 2, 3 } with addition modulo 4), and its subgroup { 0, 2 }. The quotient group **Z**_{4}/{ 0, 2 } is { { 0, 2 }, { 1, 3 } }. This is a group with identity element { 0, 2 }, and group operations such as { 0, 2 } + { 1, 3 } = { 1, 3 }. Both the subgroup { 0, 2 } and the quotient group { { 0, 2 }, { 1, 3 } } are isomorphic with **Z**_{2}.

Consider the multiplicative group . The set *N* of *n*th residues is a multiplicative subgroup isomorphic to . Then *N* is normal in *G* and the factor group *G*/*N* has the cosets *N*, (1+*n*)*N*, (1+*n*)^{2}N, ..., (1+*n*)^{n−1}N. The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of *G* without knowing the factorization of *n*.

The quotient group *G*/*G* is isomorphic to the trivial group (the group with one element), and *G*/{*e*} is isomorphic to *G*.

The order of *G*/*N*, by definition the number of elements, is equal to |*G* : *N*|, the index of *N* in *G*. If *G* is finite, the index is also equal to the order of *G* divided by the order of *N*. The set *G*/*N* may be finite, although both *G* and *N* are infinite (for example, **Z**/2**Z**).

There is a "natural" surjective group homomorphism *π* : *G* → *G*/*N*, sending each element *g* of *G* to the coset of *N* to which *g* belongs, that is: *π*(*g*) = *gN*. The mapping *π* is sometimes called the *canonical projection of G onto G/N*. Its kernel is *N*.

There is a bijective correspondence between the subgroups of *G* that contain *N* and the subgroups of *G*/*N*; if *H* is a subgroup of *G* containing *N*, then the corresponding subgroup of *G*/*N* is *π*(*H*). This correspondence holds for normal subgroups of *G* and *G*/*N* as well, and is formalized in the lattice theorem.

Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

If *G* is abelian, nilpotent, solvable, cyclic or finitely generated, then so is *G*/*N*.

If *H* is a subgroup in a finite group *G*, and the order of *H* is one half of the order of *G*, then *H* is guaranteed to be a normal subgroup, so *G*/*H* exists and is isomorphic to *C*_{2}. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if *p* is the smallest prime number dividing the order of a finite group, *G*, then if *G*/*H* has order *p*, *H* must be a normal subgroup of *G*.^{ [1] }

Given *G* and a normal subgroup *N*, then *G* is a group extension of *G*/*N* by *N*. One could ask whether this extension is trivial or split; in other words, one could ask whether *G* is a direct product or semidirect product of *N* and *G*/*N*. This is a special case of the extension problem. An example where the extension is not split is as follows: Let *G* = **Z**_{4} = {0, 1, 2, 3}, and *N* = {0, 2}, which is isomorphic to **Z**_{2}. Then *G*/*N* is also isomorphic to **Z**_{2}. But **Z**_{2} has only the trivial automorphism, so the only semi-direct product of *N* and *G*/*N* is the direct product. Since **Z**_{4} is different from **Z**_{2} × **Z**_{2}, we conclude that *G* is not a semi-direct product of *N* and *G*/*N*.

If * is a Lie group and ** is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of **, the quotient ** / ** is also a Lie group. In this case, the original group ** has the structure of a fiber bundle (specifically, a principal **-bundle), with base space ** / ** and fiber **. The dimension of ** / ** equals .*^{ [2] }

Note that the condition that * is closed is necessary. Indeed, if ** is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a Hausdorff space.*

For a non-normal Lie subgroup *, the space ** / ** of left cosets is not a group, but simply a differentiable manifold on which ** acts. The result is known as a homogeneous space.*

- ↑ Dummit & Foote (2003 , p. 120)
- ↑ John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In the mathematical field of algebraic topology, the **fundamental group** of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In mathematics, a **group** is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

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In group theory, a branch of mathematics, given a group *G* under a binary operation ∗, a subset *H* of *G* is called a **subgroup** of *G* if *H* also forms a group under the operation ∗. More precisely, *H* is a subgroup of *G* if the restriction of ∗ to *H* × *H* is a group operation on *H*. This is usually denoted *H* ≤ *G*, read as "*H* is a subgroup of *G*".

In mathematics, specifically in group theory, the concept of a **semidirect product** is a generalization of a direct product. There are two closely related concepts of semidirect product:

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 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*].

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In mathematics, more specifically in the field of group theory, a **solvable group** or **soluble group** is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

In mathematics, specifically group theory, the **index** of a subgroup *H* in a group *G* is the number of left cosets of *H* in *G*, or equivalently, the number of right cosets of *H* in *G*. The index is denoted or or . Because *G* is the disjoint union of the left cosets and because each left coset has the same size as *H*, the index is related to the orders of the two groups by the formula

In mathematics, a **free abelian group** or **free Z-module** is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In mathematics, the **modular group** is the projective special linear group PSL(2, **Z**) of 2 × 2 matrices with integer coefficients and unit determinant. The matrices *A* and −*A* are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

In mathematics, a **congruence subgroup** of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are *even*. More generally, the notion of **congruence subgroup** can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its **Whitehead torsion** which is an element in the **Whitehead group**. These concepts are named after the mathematician J. H. C. Whitehead.

In abstract algebra, a **valuation ring** is an integral domain *D* such that for every element *x* of its field of fractions *F*, at least one of *x* or *x*^{ −1} belongs to *D*.

In group theory, a field of mathematics, a **double coset** is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let *G* be a group, and let *H* and *K* be subgroups. Let *H* act on *G* by left multiplication and let *K* act on *G* by right multiplication. For each *x* in *G*, the **( H, K)-double coset of x** is the set

In additive combinatorics, **Freiman's theorem** is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if is small, then can be contained in a small generalized arithmetic progression.

In mathematics, the **Abel–Jacobi map** is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

- Dummit, David S.; Foote, Richard M. (2003),
*Abstract Algebra*(3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7 - Herstein, I. N. (1975),
*Topics in Algebra*(2nd ed.), New York: Wiley, ISBN 0-471-02371-X

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