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| {{About|order in group theory|order in other branches of mathematics|Order (mathematics)|order in other disciplines|Order (disambiguation){{!}}Order}}
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| {{Refimprove|date=May 2011}}
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| {{Group theory sidebar |Finite}}
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| In [[group theory]], a branch of [[mathematics]], the term '''order''' is used in two unrelated senses:
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| * The order of a [[group (mathematics)|group]] is its [[cardinality]], i.e., the number of elements in its [[Set (mathematics)|set]]. Also, the order, sometimes '''period''', of an [[element (group theory)|element]] ''a'' of a group is the smallest positive [[integer]] ''m'' such that ''a''<sup>''m''</sup> = ''e'' (where ''e'' denotes the [[identity element]] of the group, and ''a''<sup>''m''</sup> denotes the product of ''m'' copies of ''a''). If no such ''m'' exists, ''a'' is said to have infinite order.
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| * The total order, an antysymmetric, transitive relation ''<'' on a group with a property that whenever ''g<g' '' then ''hg<hg' '' for any ''h'' in the group. If a group admits an order it has no elements of finite order.
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| This page is about the first notion.
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| The order of a group ''G'' is denoted by ord(''G'') or |''G''| and the order of an element ''a'' is denoted by ord(''a'') or |''a''|.
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| ==Example==
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| '''Example.''' The [[symmetric group]] S<sub>3</sub> has the following multiplication table.
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| :{| cellspacing="0" cellpadding="8" border="1" | |
| |-
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| ! •
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| ! ''e'' || ''s'' || ''t'' || ''u'' || ''v'' || ''w''
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| |-
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| ! ''e''
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| | <span style="color:#009246">''e''</span> || ''s'' || ''t'' || ''u'' || ''v'' || ''w''
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| |-
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| ! ''s''
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| | ''s'' || <span style="color:#009246">''e''</span> || ''v'' || ''w'' || ''t'' || ''u''
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| |-
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| ! ''t''
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| | ''t'' || ''u'' || <span style="color:#009246">''e''</span> || ''s'' || ''w'' || ''v''
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| |-
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| ! ''u''
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| | ''u'' || ''t'' || ''w'' || <span style="color:#009246">''v''</span> || ''e'' || ''s''
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| |-
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| ! ''v''
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| | ''v'' || ''w'' || ''s'' || ''e'' || <span style="color:#009246">''u''</span> || ''t''
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| |-
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| ! ''w''
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| | ''w'' || ''v'' || ''u'' || ''t'' || ''s'' || <span style="color:#009246">''e''</span>
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| |}
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| This group has six elements, so ord(S<sub>3</sub>) = 6. By definition, the order of the identity, ''e'', is 1. Each of ''s'', ''t'', and ''w'' squares to ''e'', so these group elements have order 2. Completing the enumeration, both ''u'' and ''v'' have order 3, for ''u''<sup>2</sup> = ''v'' and ''u''<sup>3</sup> = ''vu'' = ''e'', and ''v''<sup>2</sup> = ''u'' and ''v''<sup>3</sup> = ''uv'' = ''e''.
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| ==Order and structure==
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| The order of a group and that of an element tend to speak about the structure of the group. Roughly speaking, the more complicated the factorization of the order the more complicated the group.
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| If the order of group ''G'' is 1, then the group is called a [[trivial group]]. Given an element ''a'', ord(''a'') = 1 [[if and only if]] ''a'' is the identity. If every (non-identity) element in ''G'' is the same as its inverse (so that ''a''<sup>2</sup> = ''e''), then ord(''a'') = 2 and consequently ''G'' is [[abelian group|abelian]] since <math>ab=(ab)^{-1}=b^{-1}a^{-1}=ba</math> by [[Elementary_group_theory#Inverse_of_ab|Elementary group theory]]. The converse of this statement 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:
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| :<math>2+2+2=6 \equiv 0 \pmod {6}</math>.
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| The relationship between the two concepts of order is the following: if we write
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| :<math>\langle a \rangle = \{ a^{k} : k \in \mathbb{Z} \} </math>
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| for the [[subgroup]] [[Generating set of a group|generated]] by ''a'', then
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| :<math>\operatorname{ord} (a) = \operatorname{ord}(\langle a \rangle).</math> | |
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| For any integer ''k'', we have
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| :''a''<sup>''k''</sup> = ''e'' if and only if ord(''a'') [[divisor|divides]] ''k''.
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| In general, the order of any subgroup of ''G'' divides the order of ''G''. More precisely: if ''H'' is a subgroup of ''G'', then
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| :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.)
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| 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>) = 6, the orders of the elements are 1, 2, or 3.
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| 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 journal|title=Proof of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchypf.pdf|first=Keith|last=Conrad|format=PDF|accessdate=May 14, 2011}}</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 journal|title=Consequences of Cauchy's Theorem|url=http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/cauchyapp.pdf|first=Keith|last=Conrad|format=PDF|accessdate=May 14, 2011}}</ref>
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| If ''a'' has infinite order, then all 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'':
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| :ord(''a''<sup>''k''</sup>) = ord(''a'') / [[greatest common divisor|gcd]](ord(''a''), ''k'')
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| for every integer ''k''. In particular, ''a'' and its inverse ''a''<sup>−1</sup> have the same order.
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| In any group,
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| :<math> \operatorname{ord}(ab) = \operatorname{ord}(ba)</math> | |
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| 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) = id''. 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''.
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| ==Counting by order of elements==
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| 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''), 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) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 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>.
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| ==In relation to homomorphisms==
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| [[Group homomorphism]]s tend to reduce the orders of elements: if ''f'': ''G'' → ''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'')) = ord(''a''). This can often be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism ''h'': S<sub>3</sub> → '''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.
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| ==Class equation==
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| 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:
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| :<math>|G| = |Z(G)| + \sum_{i}d_i\;</math>
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| 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>| = 1+2+3.
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| ==Open questions==
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| Several deep questions about the orders of groups and their elements are contained in the various [[Burnside problem]]s; some of these questions are still open.
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| ==References==
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| <references />
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| [[Category:Group theory]]
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