Editing Order (group theory)

{{Short description|Cardinality of a mathematical group, or of the subgroup generated by an element}}
{{About|order in group theory|other uses in mathematics|Order (mathematics)|other uses|Order (disambiguation)}}
{{For|groups with an ordering relation|partially ordered group|totally ordered group}}
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[[File:Powers of rotation, shear, and their compositions.svg|thumb|270px|Examples of [[Linear map|transformations]] with different orders: 90° rotation with order 4, [[shear mapping|shearing]] with infinite order, and their [[Function composition|compositions]] with order 3.]]
In [[mathematics]], the '''order''' of a [[finite group]] is the number of its elements. If a [[group (mathematics)|group]] is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called '''period length''' or '''period''') is the order of the subgroup generated by the element. If the group operation is denoted as a [[multiplicative group|multiplication]], the order of an element {{mvar|a}} of a group, is thus the smallest [[positive integer]] {{math|''m''}} such that {{math|1=''a''<sup>''m''</sup> = ''e''}}, where {{math|''e''}} denotes the [[identity element]] of the group, and {{math|''a''<sup>''m''</sup>}} denotes the product of {{math|''m''}} copies of {{math|''a''}}. If no such {{math|''m''}} exists, the order of {{math|''a''}} is infinite. 

The order of a group {{mvar|G}} is denoted by {{math|ord(''G'')}} or {{math|{{abs|''G''}}}}, and the order of an element {{math|''a''}} is denoted by {{math|ord(''a'')}} or {{math|{{abs|''a''}}}}, instead of <math>\operatorname{ord}(\langle a\rangle),</math> where the brackets denote the generated group.

[[Lagrange's theorem (group theory)|Lagrange's theorem]] states that for any subgroup {{math|''H''}} of a finite group {{math|''G''}}, the order of the subgroup divides the order of the group; that is, {{math|{{abs|''H''}}}} is a [[divisor]] of {{math|{{abs|''G''}}}}. In particular, the order {{math|{{abs|''a''}}}} of any element is a divisor of {{math|{{abs|''G''|}}}}.

==Example==
The [[symmetric group]] S<sub>3</sub> has the following [[Cayley table|multiplication table]].
:{| class="wikitable"
|-
!   •
! ''e'' || ''s'' || ''t'' || ''u'' || ''v'' || ''w''
|-
! ''e''
| <span style="color:#009246">''e''</span> || ''s'' || ''t'' || ''u'' || ''v'' || ''w''
|-
! ''s''
| ''s'' || <span style="color:#009246">''e''</span> || ''v'' || ''w'' || ''t'' || ''u''
|-
! ''t''
| ''t'' || ''u'' || <span style="color:#009246">''e''</span> || ''s'' || ''w'' || ''v''
|-
! ''u''
| ''u'' || ''t'' || ''w'' || <span style="color:#009246">''v''</span> || ''e'' || ''s''
|-
! ''v''
| ''v'' || ''w'' || ''s'' || ''e'' || <span style="color:#009246">''u''</span> || ''t''
|-
! ''w''
| ''w'' || ''v'' || ''u'' || ''t'' || ''s'' || <span style="color:#009246">''e''</span>
|}

This group has six elements, so {{math|1=ord(S<sub>3</sub>)&nbsp;= 6}}. By definition, the order of the identity, {{math|''e''}}, is one, since {{math|1=''e'' <sup>1</sup> = ''e''}}. Each of {{math|''s''}}, {{math|''t''}}, and {{math|''w''}} squares to {{math|''e''}}, so these group elements have order two: {{math|1={{!}}''s''{{!}} = {{!}}''t''{{!}} = {{!}}''w''{{!}} = 2}}. Finally, {{math|''u''}} and {{math|''v''}} have order 3, since {{math|1=''u''<sup>3</sup>&nbsp;= ''vu''&nbsp;= ''e''}}, and {{math|1=''v''<sup>3</sup>&nbsp;= ''uv''&nbsp;= ''e''}}.

==Order and structure==
The order of a group ''G'' and the orders of its elements give much information about the structure of the group. Roughly speaking, the more complicated the [[factorization]] of |''G''|, the more complicated the structure of ''G''.

For  |''G''| = 1, the group is [[trivial group|trivial]]. In any group, only the identity element ''a = e'' has ord(''a)'' = 1. If every non-identity element in ''G'' is equal to its inverse (so that ''a''<sup>2</sup> = ''e''), then ord(''a'') = 2; this implies ''G'' is [[abelian group|abelian]] [[group theory#Inverse of ab|since]] <math>ab=(ab)^{-1}=b^{-1}a^{-1}=ba</math>. The converse is not true; for example, the (additive) [[cyclic group]] '''Z'''<sub>6</sub> of integers [[Modular arithmetic|modulo]] 6 is abelian, but the number 2 has order 3:

:<math>2+2+2=6 \equiv 0 \pmod {6}</math>.

The relationship between the two concepts of order is the following: if we write
:<math>\langle a \rangle = \{ a^{k}\colon k \in \mathbb{Z} \} </math>
for the [[subgroup]] [[Generating set of a group|generated]] by ''a'', then
:<math>\operatorname{ord} (a) = \operatorname{ord}(\langle a \rangle).</math>

For any integer ''k'', we have
:''a<sup>k</sup>'' = ''e'' &nbsp; if and only if &nbsp; ord(''a'') [[divisor|divides]] ''k''.

In general, the order of any subgroup of ''G'' divides the order of ''G''. More precisely: if ''H'' is a subgroup of ''G'', then
:ord(''G'') / ord(''H'') = [''G'' : ''H''], where [''G'' : ''H''] is called the [[index of a subgroup|index]] of ''H'' in ''G'', an integer. This is [[Lagrange's theorem (group theory)|Lagrange's theorem]]. (This is, however, only true when G has finite order. If ord(''G'') = ∞, the quotient ord(''G'') / ord(''H'') does not make sense.)

As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group. For example, in the symmetric group shown above, where ord(S<sub>3</sub>)&nbsp;= 6, the possible orders of the elements are 1, 2, 3 or 6.

The following partial converse is true for [[finite group]]s: if ''d'' divides the order of a group ''G'' and ''d'' is a [[prime number]], then there exists an element of order ''d'' in ''G'' (this is sometimes called [[Cauchy's theorem (group theory)|Cauchy's theorem]]). The statement does not hold for [[composite number|composite]] orders, e.g. the [[Klein four-group]] does not have an element of order four. This can be shown by [[inductive proof]].<ref>{{Cite web|title=Proof of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchypf.pdf|first=Keith|last=Conrad|format=PDF|access-date=May 14, 2011|url-status=dead|archive-url=https://web.archive.org/web/20181123110229/http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchypf.pdf|archive-date=2018-11-23}}</ref> The consequences of the theorem include: the order of a group ''G'' is a power of a prime ''p'' if and only if ord(''a'') is some power of ''p'' for every ''a'' in ''G''.<ref>{{Cite web|title=Consequences of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf|first=Keith|last=Conrad|format=PDF|access-date=May 14, 2011|url-status=dead|archive-url=https://web.archive.org/web/20180712201823/http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf|archive-date=2018-07-12}}</ref>

If ''a'' has infinite order, then all non-zero powers of ''a'' have infinite order as well. If ''a'' has finite order, we have the following formula for the order of the powers of ''a'':
:ord(''a<sup>k</sup>'') = ord(''a'') / [[greatest common divisor|gcd]](ord(''a''), ''k'')<ref>Dummit, David; Foote, Richard. ''Abstract Algebra'', {{isbn|978-0471433347}}, pp. 57</ref>
for every integer ''k''. In particular, ''a'' and its inverse ''a''<sup>−1</sup> have the same order.

In any group, 
:<math> \operatorname{ord}(ab) = \operatorname{ord}(ba)</math>

There is no general formula relating the order of a product ''ab'' to the orders of ''a'' and ''b''. In fact, it is possible that both ''a'' and ''b'' have finite order while ''ab'' has infinite order, or that both ''a'' and ''b'' have infinite order while ''ab'' has finite order. An example of the former is ''a''(''x'') = 2−''x'', ''b''(''x'') = 1−''x'' with ''ab''(''x'') = ''x''−1 in the group <math>Sym(\mathbb{Z})</math>. An example of the latter is ''a''(''x'') = ''x''+1, ''b''(''x'') = ''x''−1 with ''ab''(''x'') = ''x''. If ''ab'' = ''ba'', we can at least say that ord(''ab'') divides [[least common multiple|lcm]](ord(''a''), ord(''b'')). As a consequence, one can prove that in a finite abelian group, if ''m'' denotes the maximum of all the orders of the group's elements, then every element's order divides ''m''.

==Counting by order of elements==
Suppose ''G'' is a finite group of order ''n'', and ''d'' is a divisor of ''n''.  The number of order ''d'' elements in ''G'' is a multiple of φ(''d'') (possibly zero), where φ is [[Euler's totient function]], giving the number of positive integers no larger than ''d'' and [[coprime]] to it.  For example, in the case of S<sub>3</sub>, φ(3)&nbsp;= 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2)&nbsp;= 1, and is only of limited utility for composite ''d'' such as ''d'' = 6, since φ(6) = 2, and there are zero elements of order 6 in S<sub>3</sub>.

==In relation to homomorphisms==
[[Group homomorphism]]s tend to reduce the orders of elements: if ''f'':&nbsp;''G''&nbsp;→&nbsp;''H'' is a homomorphism, and ''a'' is an element of ''G'' of finite order, then ord(''f''(''a'')) divides ord(''a''). If ''f'' is [[injective]], then ord(''f''(''a''))&nbsp;= ord(''a''). This can often be used to prove that there are no homomorphisms or no injective homomorphisms, between two explicitly given groups. (For example, there can be no nontrivial homomorphism ''h'':&nbsp;S<sub>3</sub>&nbsp;→&nbsp;'''Z'''<sub>5</sub>, because every number except zero in '''Z'''<sub>5</sub> has order 5, which does not divide the orders 1, 2, and 3 of elements in S<sub>3</sub>.) A further consequence is that [[conjugacy class|conjugate elements]] have the same order.

==Class equation<!--linked from 'Vertical bar'-->==
An important result about orders is the [[class equation]]; it relates the order of a finite group ''G'' to the order of its [[center of a group|center]] Z(''G'') and the sizes of its non-trivial [[conjugacy class]]es:
:<math>|G| = |Z(G)| + \sum_{i}d_i\;</math>
where the ''d<sub>i</sub>'' are the sizes of the non-trivial conjugacy classes; these are proper divisors of |''G''| bigger than one, and they are also equal to the indices of the centralizers  in ''G'' of the representatives of the non-trivial conjugacy classes. For example, the center of S<sub>3</sub> is just the trivial group with the single element ''e'', and the equation reads |S<sub>3</sub>|&nbsp;= 1+2+3.

==See also==
* [[Torsion subgroup]]

==Notes==
{{reflist|30em}}

==References==
* Dummit, David; Foote, Richard. Abstract Algebra, {{ISBN|978-0471433347}}, pp. 20, 54–59, 90
* Artin, Michael. Algebra, {{ISBN|0-13-004763-5}}, pp. 46–47

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[[Category:Group theory]]
[[Category:Algebraic properties of elements]]