Quotient group

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.

For a congruence relation on 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 Template:Tmath, where $G$ is the original group and $N$ is the normal subgroup. This is read as 'Template:Tmath', where $\text{mod}$ is short for modulo. (The notation Template:Tmath should be interpreted with caution, as some authors (e.g., Vinberg) use it to represent the left cosets of $H$ in $G$ for any subgroup $H$, even though these cosets do not form a group if $H$ is not normal in Template:Tmath. Others (e.g., Dummit and Foote) use this notation to refer only to the quotient group, with the appearance of this notation implying that $H$ is normal in Template:Tmath.)

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 Template:Tmath. Specifically, the image of $G$ under a homomorphism $\phi :G\to H$ is isomorphic to $G/\ker (\phi )$ where $\ker (\phi )$ denotes the kernel of Template:Tmath.

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.

Definition and illustration

Given a group $G$ and a subgroup Template:Tmath, and a fixed element $a\in G$, one can consider the corresponding left coset: Template:Tmath. 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: Template:Tmath, which are the even integers, and Template:Tmath, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).

For a general subgroup Template:Tmath, it is desirable to define a compatible group operation on the set of all possible cosets, Template:Tmath. 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 Template:Tmath. A normal subgroup of $G$ is denoted Template:Tmath.

Definition

Let $N$ be a normal subgroup of a group Template:Tmath. Define the set $G/N$ to be the set of all left cosets of $N$ in Template:Tmath. That is, Template:Tmath.

Since the identity element Template:Tmath, Template:Tmath. Define a binary operation on the set of cosets, Template:Tmath, as follows. For each $aN$ and $bN$ in Template:Tmath, the product of $aN$ and Template:Tmath, Template:Tmath, is Template:Tmath. This works only because $(ab)N$ does not depend on the choice of the representatives, $a$ and Template:Tmath, of each left coset, $aN$ and Template:Tmath. To prove this, suppose $xN=aN$ and $yN=bN$ for some Template:Tmath. 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 Template:Tmath is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on Template:Tmath.

To show that it is necessary, consider that for a subgroup $N$ of Template:Tmath, we have been given that the operation is well defined. That is, for all $xN=aN$ and Template:Tmath for Template:Tmath.

Let $n\in N$ and Template:Tmath. Since Template:Tmath, we have Template:Tmath.

Now, $gN=(ng)N\iff N=(g^{-1}ng)N\iff g^{-1}ng\in N,\mathrm{\forall }n\in N$ and Template:Tmath.

Hence $N$ is a normal subgroup of Template:Tmath.

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

Due to the normality of Template:Tmath, 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 Template:Tmath.

Example: Addition modulo 6

For example, consider the group with addition modulo 6: Template:Tmath. Consider the subgroup Template:Tmath, which is normal because $G$ is abelian. Then the set of (left) cosets is of size three:

$G/N=\{a+N:a\in G\}=\{\{0,3\},\{1,4\},\{2,5\}\}=\{0+N,1+N,2+N\}.$

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.

Motivation for the name "quotient"

The quotient group $G/N$ can be compared to division of integers. When dividing 12 by 3 one obtains the result 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although one ends up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects: in the quotient Template:Tmath, the group structure is used to form a natural "regrouping". These are the cosets of $N$ in Template:Tmath. 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.

Examples

Even and odd integers

Consider the group of integers $\mathbb{Z}$ (under addition) and the subgroup $2\mathbb{Z}$ consisting of all even integers. This is a normal subgroup, because $\mathbb{Z}$ is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group $\mathbb{Z}/2\mathbb{Z}$ 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 $\mathbb{Z}/2\mathbb{Z}$ equals the set $\{0,1\}$ with addition modulo 2.

Example further explained...

Let $\gamma (m)$ be the remainders of $m\in \mathbb{Z}$ when dividing by Template:Tmath. Then, $\gamma (m)=0$ when $m$ is even and $\gamma (m)=1$ when $m$ is odd.

By definition of Template:Tmath, the kernel of Template:Tmath, Template:Tmath, is the set of all even integers.

Let Template:Tmath. Then, $H$ is a subgroup, because the identity in Template:Tmath, which is Template:Tmath, is in Template:Tmath, the sum of two even integers is even and hence if $m$ and $n$ are in Template:Tmath, $m+n$ is in $H$ (closure) and if $m$ is even, $-m$ is also even and so $H$ contains its inverses.

Define $\mu :\mathbb{Z}/H\to {\mathrm{Z}}_2$ as $\mu (aH)=\gamma (a)$ for $a\in \mathbb{Z}$ and $\mathbb{Z}/H$ is the quotient group of left cosets; Template:Tmath.

Note that we have defined Template:Tmath, $\mu (aH)$ is $1$ if $a$ is odd and $0$ if $a$ is even.

Thus, $\mu$ is an isomorphism from $\mathbb{Z}/H$ to Template:Tmath.

Remainders of integer division

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

Complex integer roots of 1

The cosets of the fourth roots of unity N in the twelfth roots of unity G.

The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group Template:Tmath, 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.

Real numbers modulo the integers

Consider the group of real numbers $ℝ$ under addition, and the subgroup $\mathbb{Z}$ of integers. Each coset of $\mathbb{Z}$ in $ℝ$ is a set of the form Template:Tmath, where $a$ is a real number. Since $a_1+\mathbb{Z}$ and $a_2+\mathbb{Z}$ are identical sets when the non-integer parts of $a_1$ and $a_2$ are equal, one may impose the restriction $0\le a<1$ without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group $ℝ/\mathbb{Z}$ is isomorphic to the circle group, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, that is, the special orthogonal group Template:Tmath. An isomorphism is given by $f(a+\mathbb{Z})=\exp (2\pi ia)$ (see Euler's identity).

Matrices of real numbers

If $G$ is the group of invertible $3\times 3$ real matrices, and $N$ is the subgroup of $3\times 3$ 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 Template:Tmath.

Integer modular arithmetic

Consider the abelian group ${\mathrm{Z}}_4=\mathbb{Z}/4\mathbb{Z}$ (that is, the set $\{0,1,2,3\}$ with addition modulo 4), and its subgroup Template:Tmath. The quotient group ${\mathrm{Z}}_4/\{0,2\}$ is Template:Tmath. This is a group with identity element Template:Tmath, and group operations such as Template:Tmath. Both the subgroup $\{0,2\}$ and the quotient group $\{\{0,2\},\{1,3\}\}$ are isomorphic with Template:Tmath.

Integer multiplication

Consider the multiplicative group Template:Tmath. The set $N$ of Template:Tmathth residues is a multiplicative subgroup isomorphic to Template:Tmath. Then $N$ is normal in $G$ and the factor group $G/N$ has the cosets Template:Tmath. 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 Template:Tmath.

Properties

The quotient group $G/G$ is isomorphic to the trivial group (the group with one element), and $G/\left\{e\right\}$ is isomorphic to Template:Tmath.

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

There is a "natural" surjective group homomorphism Template:Tmath, sending each element $g$ of $G$ to the coset of $N$ to which $g$ belongs, that is: Template:Tmath. The mapping $\pi$ is sometimes called the canonical projection of $G$ onto Template:Tmath. Its kernel is Template:Tmath.

There is a bijective correspondence between the subgroups of $G$ that contain $N$ and the subgroups of Template:Tmath; if $H$ is a subgroup of $G$ containing Template:Tmath, then the corresponding subgroup of $G/N$ is Template:Tmath. 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 Template:Tmath.

If $H$ is a subgroup in a finite group Template:Tmath, and the order of $H$ is one half of the order of Template:Tmath, then $H$ is guaranteed to be a normal subgroup, so $G/H$ exists and is isomorphic to Template:Tmath. 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, Template:Tmath, then if $G/H$ has order Template:Tmath, $H$ must be a normal subgroup of Template:Tmath.

Given $G$ and a normal subgroup Template:Tmath, then $G$ is a group extension of $G/N$ by Template:Tmath. 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 Template:Tmath. This is a special case of the extension problem. An example where the extension is not split is as follows: Let Template:Tmath, and Template:Tmath, which is isomorphic to Template:Tmath. Then $G/N$ is also isomorphic to Template:Tmath. But ${\mathrm{Z}}_2$ has only the trivial automorphism, so the only semi-direct product of $N$ and $G/N$ is the direct product. Since ${\mathrm{Z}}_4$ is different from Template:Tmath, we conclude that $G$ is not a semi-direct product of $N$ and Template:Tmath.

Quotients of Lie groups

If $G$ is a Lie group and $N$ is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of Template:Tmath, the quotient $G/N$ is also a Lie group. In this case, the original group $G$ has the structure of a fiber bundle (specifically, a [[principal bundle|principal Template:Tmath-bundle]]), with base space $G/N$ and fiber Template:Tmath. The dimension of $G/N$ equals Template:Tmath.

Note that the condition that $N$ is closed is necessary. Indeed, if $N$ 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 Template:Tmath, the space $G/N$ of left cosets is not a group, but simply a differentiable manifold on which $G$ acts. The result is known as a homogeneous space.

See also

• Group extension
• Quotient category
• Short exact sequence

References

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