In group theory, a branch of mathematics, the term order is used in two closely related senses: Group theory is that branch of mathematics concerned with the study of groups. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

• the order of a group is its cardinality, i.e. the number of its elements;
• the order, sometimes period, of an element a of a group is the smallest positive integer m such that a^m = e (where e denotes the identity element of the group, and a^m denotes the product of m copies of a). If no such m exists, we say that a has infinite order.

We denote the order of a group G by ord(G) or |G| and the order of an element a by ord(a) or |a|. This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... The integers are commonly denoted by the above symbol. ... For other uses, see identity (disambiguation). ...

Example

Example. The symmetric group S₃ has the following multiplication table. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

•	e	s	t	u	v	w
e	e	s	t	u	v	w
s	s	e	v	w	t	u
t	t	u	e	s	w	v
u	u	t	w	v	e	s
v	v	w	s	e	u	t
w	w	v	u	t	s	e

This group has six elements, so ord(S₃) = 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 u² = v and u³ = vu = e, and v² = u and v³ = uv = e.

Order and structure

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.

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 a² = e), then ord(a) = 2 and consequently G is abelian since ab = (bb)ab(aa) = b(ba)(ba)a = ba. The converse of this statement is not true; for example, the (additive) cyclic group Z₆ of integers modulo 6 is abelian, but the number 2 has order 3 (2+2+2 = 6 ≡ 0 (mod 6)). The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of... The integers are commonly denoted by the above symbol. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ...

The relationship between the two concepts of order is the following: if we write

for the subgroup generated by a, then In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

For any integer k, we have

a^k = e   if and only if   ord(a) 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

ord(G) / ord(H) = [G : H],

where [G : H] is the index of H in G, an integer. This is Lagrange's theorem. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ...

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(S₃) = 6, the orders of the elements are 1, 2, or 3.

The following partial converse is true for finite groups: 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). The statement does not hold for composite orders, e.g. the Klein four-group does not have an element of order four). This can be shown by inductive proof [1]. 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 [2]. In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... -1... Look up composite in Wiktionary, the free dictionary. ... This article is about the mathematical group. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

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:

ord(a^k) = ord(a) / gcd(ord(a), k)

for every integer k. In particular, a and its inverse a⁻¹ have the same order. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

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. If ab = ba, we can at least say that ord(ab) divides 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. In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...

Counting by order of elements

Suppose G is a finite group of order n, and d is a divisor of n. The number of elements in G of order d 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 S₃, 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 S₃. The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...

In relation to homomorphisms

Group homomorphisms 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: S₃ → Z₅, because every number but zero in Z₅ has order 5, which does not divide the orders 1, 2, and 3 of elements in S₃.) A further consequence is that conjugate elements have the same order. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

Class equation

An important result about orders is the class equation; it relates the order of a finite group G to the order of its center Z(G) and the sizes of its non-trivial conjugacy classes: In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of... In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

where the d_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 S₃ is just the trivial group with the single element e, and the equation reads |S₃| = 1+2+3.

Open questions

Several deep questions about the orders of groups and their elements are contained in the various Burnside problems; some of these questions are still open. One of the oldest open problems in group theory was first posed by William Burnside in a paper published in 1902. ...