In

_{3} has the following

^{2} = ''e''), then ord(''a'') = 2; this implies ''G'' is abelian since $ab=(ab)^=b^a^=ba$. The converse is not true; for example, the (additive) _{6} of integers modulo 6 is abelian, but the number 2 has order 3:
:$2+2+2=6\; \backslash equiv\; 0\; \backslash pmod$.
The relationship between the two concepts of order is the following: if we write
:$\backslash langle\; a\; \backslash rangle\; =\; \backslash $
for the ^{k}'' = ''e'' if and only if ord(''a'') _{3}) = 6, the possible orders of the elements are 1, 2, 3 or 6.
The following partial converse is true for ^{k}'') = ord(''a'') / (ord(''a''), ''k'')Dummit, David; Foote, Richard. ''Abstract Algebra'', , pp. 57
for every integer ''k''. In particular, ''a'' and its inverse ''a''^{−1} have the same order.
In any group,
:$\backslash operatorname(ab)\; =\; \backslash operatorname(ba)$
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 $Sym(\backslash mathbb)$. 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 (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''.

_{3}, 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_{3}.

_{3} → Z_{5}, because every number except zero in Z_{5} has order 5, which does not divide the orders 1, 2, and 3 of elements in S_{3}.) A further consequence is that

_{i}'' 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_{3} is just the trivial group with the single element ''e'', and the equation reads , S_{3}, = 1+2+3.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the order of a finite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

is the number of its elements. If a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

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 multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

, the order of an element of a group, is thus the smallest positive integer
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

such that , where denotes the identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

of the group, and denotes the product of copies of . If no such exists, the order of is infinite.
The order of a group is denoted by or , and the order of an element is denoted by or , instead of $\backslash operatorname(\backslash langle\; a\backslash rangle),$ where the brackets denote the generated group.
Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of . In particular, the order of any element is a divisor of .
Example

Thesymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

Smultiplication table
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
:
This group has six elements, so . By definition, the order of the identity, , is one, since . Each of , , and squares to , so these group elements have order two: . Finally, and have order 3, since , and .
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 thefactorization
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of , ''G'', , the more complicated the structure of ''G''.
For , ''G'', = 1, the group is 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''cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

Zsubgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

generated by ''a'', then
:$\backslash operatorname\; (a)\; =\; \backslash operatorname(\backslash langle\; a\; \backslash rangle).$
For any integer ''k'', we have
:''adivides
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''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
Index may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastructure in the ''Halo'' series ...

of ''H'' in ''G'', an integer. This is 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(Sfinite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s: if ''d'' divides the order of a group ''G'' and ''d'' is a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

, then there exists an element of order ''d'' in ''G'' (this is sometimes called Cauchy's theorem). The statement does not hold for composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
* ...

orders, e.g. the Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...

does not have an element of order four). This can be shown by inductive proof. 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''.
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(''aCounting 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 isEuler's totient function
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and numbe ...

, giving the number of positive integers no larger than ''d'' and coprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

to it. For example, in the case of SIn relation to homomorphisms

Group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, 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'': Sconjugate elements
In mathematics, in particular field theory (mathematics), field theory, the conjugate elements of an algebraic element ''α'', over a field extension ''L''/''K'', are the roots of the minimal polynomial (field theory), minimal polynomial ''p'' ...

have the same order.
Class equation

An important result about orders is the class equation; it relates the order of a finite group ''G'' to the order of itscenter
Center or centre may refer to:
Mathematics
*Center (geometry)
In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...

Z(''G'') and the sizes of its non-trivial conjugacy class
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

es:
:$,\; G,\; =\; ,\; Z(G),\; +\; \backslash sum\_d\_i\backslash ;$
where the ''dSee also

*Torsion subgroupIn the theory of abelian group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (math ...

Notes

References

* Dummit, David; Foote, Richard. Abstract Algebra, , pp. 20, 54–59, 90 * Artin, Michael. Algebra, , pp. 46–47 {{Authority control Group theory Algebraic properties of elements