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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G} is the original group and $N$ is the normal subgroup. This is read as 'Template:Tmath', where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varphi :G\rightarrow H} is isomorphic to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G\,/\,\ker(\varphi )} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ker(\varphi )} 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.

For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.

Definition and illustration

Given a group $G$ and a subgroup Template:Tmath, and a fixed element Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle aN} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle bN} in Template:Tmath, the product of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle aN} and Template:Tmath, Template:Tmath, is Template:Tmath. This works only because Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (ab)N} does not depend on the choice of the representatives, $a$ and Template:Tmath, of each left coset, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle aN} and Template:Tmath. To prove this, suppose Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle xN=aN} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle yN=bN} for some Template:Tmath. Then

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle (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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle xN=aN} and Template:Tmath for Template:Tmath.

Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n\in N} and Template:Tmath. Since Template:Tmath, we have Template:Tmath.

Now, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle gN=(ng)N\Leftrightarrow N=(g^{-1}ng)N\Leftrightarrow g^{-1}ng\in N,\;\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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G\,/\,N=\left\{a+N:a\in G\right\}=\left\{\left\{0,3\right\},\left\{1,4\right\},\left\{2,5\right\}\right\}=\left\{0+N,1+N,2+N\right\}.}

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {Z} \,/\,2\mathbb {Z} } is the cyclic group with two elements. This quotient group is isomorphic with the set $\left\{0,1\right\}$ with addition modulo 2; informally, it is sometimes said that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {Z} \,/\,2\mathbb {Z} } equals the set $\left\{0,1\right\}$ with addition modulo 2.

Example further explained...

Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma (m)} be the remainders of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle m\in \mathbb {Z} } when dividing by Template:Tmath. Then, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma (m)=0} when $m$ is even and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n } are in Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m+n } is in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H } (closure) and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } is even, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -m } is also even and so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H } contains its inverses.

Define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu : \mathbb{Z} / H \to \mathrm{Z}_2 } as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(aH)=\gamma(a) } for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in\Z } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} / H} is the quotient group of left cosets; Template:Tmath.

Note that we have defined Template:Tmath, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(aH) } is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 } if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is odd and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is even.

Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } is an isomorphism from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z} / H} to Template:Tmath.

Remainders of integer division

A slight generalization of the last example. Once again consider the group of integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} under addition. Let Template:Tmath be any positive integer. We will consider the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\Z} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} consisting of all multiples of Template:Tmath. Once again Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\Z} is normal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} is abelian. The cosets are the collection Template:Tmath. An integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} belongs to the coset Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the remainder when dividing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} by Template:Tmath. The quotient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z\,/\,n\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

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} under addition, and the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} of integers. Each coset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} is a set of the form Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is a real number. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1+\Z} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2+\Z} are identical sets when the non-integer parts of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2} are equal, one may impose the restriction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R\,/\,\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a+\Z) = \exp(2\pi ia)} (see Euler's identity).

Matrices of real numbers

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is the group of invertible Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 3} real matrices, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \times 3} real matrices with determinant 1, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is normal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (since it is the kernel of the determinant homomorphism). The cosets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} are the sets of matrices with a given determinant, and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,/\,N} is isomorphic to the multiplicative group of non-zero real numbers. The group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is known as the special linear group Template:Tmath.

Integer modular arithmetic

Consider the abelian group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Z}_4 = \Z\,/\,4 \Z} (that is, the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{0, 1, 2, 3 \right\}} with addition modulo 4), and its subgroup Template:Tmath. The quotient group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Z}_4\,/\,\left\{0, 2\right\}} is Template:Tmath. This is a group with identity element Template:Tmath, and group operations such as Template:Tmath. Both the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{0, 2\right\}} and the quotient group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{\left\{ 0, 2 \right\}, \left\{1, 3 \right\} \right\}} are isomorphic with Template:Tmath.

Integer multiplication

Consider the multiplicative group Template:Tmath. The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} of Template:Tmathth residues is a multiplicative subgroup isomorphic to Template:Tmath. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is normal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and the factor group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} without knowing the factorization of Template:Tmath.

Properties

The quotient group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,/\,G} is isomorphic to the trivial group (the group with one element), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} in Template:Tmath. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is finite, the index is also equal to the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} divided by the order of Template:Tmath. The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,/\,N} may be finite, although both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} are infinite (for example, Template:Tmath).

There is a "natural" surjective group homomorphism Template:Tmath, sending each element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} to the coset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} to which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} belongs, that is: Template:Tmath. The mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} is sometimes called the canonical projection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} onto Template:Tmath. Its kernel is Template:Tmath.

There is a bijective correspondence between the subgroups of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} that contain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} and the subgroups of Template:Tmath; if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is a subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} containing Template:Tmath, then the corresponding subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,/\,N} is Template:Tmath. This correspondence holds for normal subgroups of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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

• 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