Order (group theory)

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Order (group theory)

In group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
, a branch of mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the term order is used in two closely related senses:

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 group has six elements, so ord(S₃) = 6.

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In group theory

Group theory

Mathematics

the order of a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
is its cardinality
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
, i.e. the number of its elements;
the order, sometimes period, of an element a of a group is the smallest positive integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
m such that a^m = e (where e denotes the identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
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. All elements of finite groups have finite 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|.

Example

Example. The symmetric group

Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
S₃ has the following multiplication table.

•	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

Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group....
. Given an element a, ord(a) = 1 if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
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

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
since ab = (bb)ab(aa) = b(ba)(ba)a = ba. The converse of this statement is not true; for example, the (additive) cyclic group

Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
Z₆ of integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s modulo

Modulo

The word modulo, in the mathematical community, is often used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....
6 is abelian, but the number 2 has order 3 (2+2+2 = 6 ≡ 0 (mod 6)).

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

Subgroup

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 *....
generated by a, then

For any integer k, we have

a^k = e if and only if ord(a) divides

Divisor

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....
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 the index of H in G, an integer. This is Lagrange's theorem

Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G....
.

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 group

Finite group

In mathematics, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
s: if d divides the order of a group G and d is a prime number

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
, then there exists an element of order d in G (this is sometimes called Cauchy's theorem

Cauchy's theorem (group theory)

Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the Order of G , then G contains an element of order p....
). The statement does not hold for composite

Composite number

A composite number is a negative and non-negative numbers integer which has a positive divisor other than one or itself. In other words, if 0 < n is an integer and there are integers 1 < a, b < n such that n = a ? b then n is composite....
orders, e.g. the Klein four-group

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of Order 2 ....
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 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

Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
(ord(a), k)

for every integer k. In particular, a and its inverse a^-1 have the same order.

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

Least common multiple

In arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b....
(ord(a), ord(b)). As a consequence, one can prove that in a finite abelian group

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
, 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 elements in G of order d is a multiple of φ(d), where φ is Euler's totient function

Euler's totient function

In number theory, the totient of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n....
, giving the number of positive integers no larger than d and coprime

Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common divisor other than 1 or, equivalently, if their greatest common divisor is 1....
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.

In relation to homomorphisms

Group homomorphism

In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
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: 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

Conjugacy class

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 class

Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set 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....
es: 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.