|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Group theory sidebar |Basics}}
| | Interior Designer Joseph from Saint-Augustin, enjoys butterflies, 10 hatha yoga positions and sky diving. Enjoys travel and was motivated after gonna Laponian Area. |
| | |
| In [[mathematics]], specifically [[group theory]], a '''quotient group''' (or '''factor group''') is a group obtained by aggregating similar elements of a larger group using an [[equivalence relation]]. For example, the [[cyclic group]] of [[modular arithmetic|addition modulo ''n'']] can be obtained from the [[integer]]s 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.
| |
| | |
| 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 [[coset]]s of that normal subgroup. The resulting quotient is written '''{{nowrap|''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 [[modular arithmetic|modulo]].)
| |
| | |
| Much of the importance of quotient groups is derived from their relation to [[group homomorphism|homomorphisms]]. The [[isomorphism theorem#First isomorphism theorem|first isomorphism theorem]] states that the [[image (mathematics)|image]] of any group ''G'' under a homomorphism is always [[group isomorphism|isomorphic]] to a quotient of ''G''. Specifically, the image of ''G'' under a homomorphism {{nowrap|''φ'': ''G'' → ''H''}} is isomorphic to {{nowrap|''G'' / ker(''φ'')}} where ker(''φ'') denotes the [[kernel (algebra)#Group homomorphisms|kernel]] of ''φ''.
| |
| | |
| The [[duality (mathematics)|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 object]]s, which are [[dual (category theory)|dual]] to [[subobject]]s. For other examples of quotient objects, see [[quotient ring]], [[quotient space (linear algebra)]], [[quotient space|quotient space (topology)]], and [[quotient set]].
| |
| | |
| ==Product of subsets of a group==
| |
| {{main|Product of group subsets}}
| |
| In the following discussion, we will use a binary operation on the ''subsets'' of ''G'': if two subsets ''S'' and ''T'' of ''G'' are given, we define their product as {{nowrap|1=''ST'' = {''st'' : ''s'' ∈ ''S'' ∧ ''t'' ∈ ''T''} }}. This operation is [[associative]] and has as [[identity element]] the [[Singleton (mathematics)|singleton]] {''e''}, where ''e'' is the identity element of ''G''. Thus, the set of all subsets of ''G'' forms a [[monoid]] under this operation.
| |
| | |
| In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is:
| |
| | |
| :''A quotient group of a group ''G'' is a [[Partition of a set|partition]] of ''G'' which is itself a group under this operation''.
| |
| | |
| It is fully determined by the subset containing ''e''. A [[normal subgroup]] of ''G'' is the set containing ''e'' in any such partition. The subsets in the partition are the [[coset]]s of this normal subgroup.
| |
| | |
| A subgroup ''N'' of a group ''G'' is normal [[if and only if]] the coset equality ''aN'' = ''Na'' holds for all ''a'' in ''G''. In terms of the binary operation on subsets defined above, a normal subgroup of ''G'' is a subgroup that commutes with every subset of ''G'' and is denoted {{nowrap|''N'' ◁ ''G''}}. A subgroup that permutes with every subgroup of ''G'' is called a [[permutable subgroup]].
| |
| | |
| ==Definition==
| |
| Let ''N'' be a [[normal subgroup]] of a group ''G''. We define the set ''G''/''N'' to be the set of all left [[coset]]s of ''N'' in ''G'', i.e., {{nowrap|1=''G''/''N'' = { ''aN'' : ''a'' in ''G'' } }}. The group operation on ''G''/''N'' is the product of subsets defined above. In other words, for each ''aN'' and ''bN'' in ''G''/''N'', the product of ''aN'' and ''bN'' is (''aN'')(''bN''). This operation is closed, because (''aN'')(''bN'') really is a left coset:
| |
| | |
| :(''aN'')(''bN'') = ''a''(''Nb'')''N'' = ''a''(''bN'')''N'' = (''ab'')''NN'' = (''ab'')''N''.
| |
| | |
| The normality of ''N'' is used in this equation. Because of the normality of ''N'', the left cosets and right cosets of ''N'' in ''G'' are equal, and so ''G''/''N'' could be defined as the set of right cosets of ''N'' in ''G''. Because the operation is derived from the product of subsets of ''G'', the operation is [[well-defined]] (does not depend on the particular choice of representatives), associative, and has identity element ''N''. The inverse of an element ''aN'' of ''G''/''N'' is ''a''<sup>−1</sup>''N''.
| |
| | |
| For example, consider the group with addition modulo 6:
| |
| | |
| : ''G'' = {0, 1, 2, 3, 4, 5}.
| |
| | |
| Let
| |
| | |
| : ''N'' = {0, 3}.
| |
| | |
| The quotient group is:
| |
| | |
| : ''G''/''N'' = { ''aN'' : ''a'' ∈ G } = { ''a''{0, 3} : ''a'' ∈ {0, 1, 2, 3, 4, 5} } =
| |
| :: { 0{0, 3}, 1{0, 3}, 2{0, 3}, 3{0, 3}, 4{0, 3}, 5{0, 3} } =
| |
| :: { {(0+0) mod 6, (0+3) mod 6}, {(1+0) mod 6, (1+3) mod 6},
| |
| ::: {(2+0) mod 6, (2+3) mod 6}, {(3+0) mod 6, (3+3) mod 6},
| |
| ::: {(4+0) mod 6, (4+3) mod 6}, {(5+0) mod 6, (5+3) mod 6} } =
| |
| :: { {0, 3}, {1, 4}, {2, 5}, {3, 0}, {4, 1}, {5, 2} } =
| |
| :: { {0, 3}, {1, 4}, {2, 5}, {0, 3}, {1, 4}, {2, 5} } =
| |
| :: { {0, 3}, {1, 4}, {2, 5} }.
| |
| | |
| The basic argument above is still valid if ''G''/''N'' is defined to be the set of all right [[coset]]s.
| |
| | |
| ==Motivation for the name "quotient"==
| |
| The reason ''G''/''N'' is called a quotient group comes from [[division (mathematics)|division]] of [[integer]]s. 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, however 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.
| |
| | |
| ==Examples==
| |
| [[File:Normal subgroup illustration.png|right|thumb|The cosets of the fourth [[roots of unity]] ''N'' in the twelfth roots of unity ''G''.]]
| |
| | |
| Consider the group of [[integer]]s '''Z''' (under addition) and the subgroup 2'''Z''' consisting of all even integers. This is a normal subgroup, because '''Z''' is [[abelian group|abelian]]. There are only two cosets: the set of even integers and the set of odd integers; therefore, the quotient group '''Z'''/2'''Z''' is the cyclic group with two elements. This quotient group is isomorphic with the set {{nowrap|{ 0, 1 } }} with addition modulo 2; informally, it is sometimes said that '''Z'''/2'''Z''' ''equals'' the set {{nowrap|{ 0, 1 } }} with addition modulo 2.
| |
| | |
| 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 [[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 number]]s '''R''' under addition, and the subgroup '''Z''' of integers. The cosets of '''Z''' in '''R''' are all sets of the form ''a''+'''Z''', with {{nowrap|0 ≤ ''a'' < 1}} a real number. 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'''/'''Z''' is isomorphic to the [[circle group]] S<sup>1</sup>, the group of [[complex number]]s of [[absolute value]] 1 under multiplication, or correspondingly, the group of [[rotation]]s in 2D about the origin, i.e., the special [[orthogonal group]] SO(2). An isomorphism is given by {{nowrap|1=''f''(''a''+'''Z''') = exp(2''πia'')}} (see [[Euler's identity]]).
| |
| | |
| If ''G'' is the group of invertible 3 × 3 real [[matrix (mathematics)|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 (algebra)|kernel]] of the determinant [[group homomorphism|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.
| |
| | |
| Consider the abelian group {{nowrap|1='''Z'''<sub>4</sub> = '''Z'''/4'''Z'''}} (that is, the set {{nowrap|{ 0, 1, 2, 3 } }} with addition [[Modular arithmetic|modulo]] 4), and its subgroup {{nowrap|{ 0, 2 } }}. The quotient group {{nowrap|'''Z'''<sub>4</sub>/{ 0, 2 } }} is {{nowrap|{ { 0, 2 }, { 1, 3 } } }}. This is a group with identity element {{nowrap|{ 0, 2 } }}, and group operations such as {{nowrap|1={ 0, 2 } + { 1, 3 } = { 1, 3 }}}. Both the subgroup {{nowrap|{ 0, 2 } }} and the quotient group {{nowrap|{ { 0, 2 }, { 1, 3 } } }} are isomorphic with '''Z'''<sub>2</sub>.
| |
| | |
| Consider the multiplicative group <math>G=\mathbf{Z}^*_{n^2}</math>. The set ''N'' of ''n''th residues is a multiplicative subgroup isomorphic to <math>\mathbf{Z}^*_{n}</math>. Then ''N'' is normal in ''G'' and the factor group ''G''/''N'' has the cosets ''N'', (1+''n'')''N'', (1+''n'')<sup>2</sup>N, ..., (1+''n'')<sup>''n''−1</sup>N. The [[Pallier 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''.
| |
| | |
| ==Properties==
| |
| The quotient group {{nowrap|''G'' / ''G''}} is [[group isomorphism|isomorphic]] to the [[Trivial (mathematics)|trivial group]] (the group with one element), and {{nowrap|''G'' / {''e''} }} is isomorphic to ''G''.
| |
| | |
| The [[group order|order]] of {{nowrap|''G'' / ''N''}}, by definition the number of elements, is equal to {{nowrap|{{!}}''G'' : ''N''{{!}}}}, the [[index of a subgroup|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''. Note that {{nowrap|''G'' / ''N''}} may be finite, although both ''G'' and ''N'' are infinite (e.g. {{nowrap|'''Z''' / 2'''Z'''}}).
| |
| | |
| There is a "natural" [[surjective]] [[group homomorphism]] {{nowrap|''π'' : ''G'' → ''G'' / ''N''}}, sending each element ''g'' of ''G'' to the coset of ''N'' to which ''g'' belongs, that is: {{nowrap|1=''π''(''g'') = ''gN''}}. The mapping ''π'' is sometimes called the ''canonical projection of G onto {{nowrap|G / N}}''. Its [[kernel (algebra)|kernel]] is ''N''.
| |
| | |
| There is a bijective correspondence between the subgroups of ''G'' that contain ''N'' and the subgroups of {{nowrap|''G'' / ''N''}}; if ''H'' is a subgroup of ''G'' containing ''N'', then the corresponding subgroup of {{nowrap|''G'' / ''N''}} is ''π''(''H''). This correspondence holds for normal subgroups of ''G'' and {{nowrap|''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 theorem]]s.
| |
| | |
| If ''G'' is [[abelian group|abelian]], [[nilpotent group|nilpotent]] or [[solvable group|solvable]], then so is {{nowrap|''G'' / ''N''}}.
| |
| | |
| If ''G'' is [[cyclic group|cyclic]] or [[generating set of a group|finitely generated]], then so is {{nowrap|''G'' / ''N''}}.
| |
| | |
| If ''N'' is contained in the [[center (group theory)|center]] of ''G'', then ''G'' is called the [[Group extension%23Central extension|central extension]] of the quotient group.
| |
| | |
| 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 {{nowrap|''G'' / ''H''}} exists and is isomorphic to ''C''<sub>2</sub>. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups.
| |
| | |
| To extend the above, if ''p'' is the smallest prime number dividing the order of a finite group, ''G'', then if {{nowrap|''G'' / ''H''}} has order ''p'', ''H'' must be a normal subgroup of ''G''.<ref>{{harvtxt|Dummit|Foote|2003|p=120}}</ref>
| |
| | |
| Every finitely generated group is isomorphic to a quotient of a [[free group]].
| |
| | |
| Sometimes, but not necessarily, a group ''G'' can be reconstructed from {{nowrap|''G'' / ''N''}} and ''N'', as a [[direct product of groups|direct product]] or [[semidirect product]]. The problem of determining when this is the case is known as the [[extension problem]]. An example where it is ''not'' possible is as follows. {{nowrap|'''Z'''<sub>4</sub> / { 0, 2 } }} is isomorphic to '''Z'''<sub>2</sub>, and {{nowrap|{ 0, 2 } }} also, but the only semidirect product is the direct product, because '''Z'''<sub>2</sub> has only the trivial [[automorphism]]. Therefore '''Z'''<sub>4</sub>, which is different from {{nowrap|'''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>}}, cannot be reconstructed.
| |
| | |
| ==Quotients of Lie groups==
| |
| If ''G'' is a [[Lie group]] and ''N'' is a normal [[Lie subgroup]] of ''G'', the quotient {{nowrap|''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 ''N''-bundle]]), with base space {{nowrap|''G'' / ''N''}} and fiber ''N''.
| |
| | |
| For a non-normal Lie subgroup ''N'', the space {{nowrap|''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==
| |
| | |
| *[[Quotient ring]], also called a ''factor ring''
| |
| *[[Group extension]]
| |
| *[[Extension problem]]
| |
| *[[Lattice theorem]]
| |
| *[[Quotient category]]
| |
| *[[Short exact sequence]]
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{citation | last1 = Dummit | first1 = David S. | last2 = Foote | first2 = Richard M. | title = Abstract Algebra | year = 2004 | edition = 3rd | publisher = [[John Wiley and Sons|Wiley]] | location = New York | isbn = 978-0-471-43334-7 }}
| |
| * {{citation | first1 = I.N. | last1 = Herstein | year = 1975 | title = Topics in Algebra | edition = 2nd | publisher = [[John Wiley and Sons|Wiley]] | location = New York | isbn = 0-471-02371-X }}
| |
| | |
| [[Category:Group theory]]
| |