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In group theory, a branch of mathematics, the term order is used in two closely related senses: Wikipedia does not have an article with this exact name. ... In number theory, given an integer a and a positive integer n with gcd(a,n) = 1, the multiplicative order of a modulo n is the smallest positive integer k with ak ≡ 1 (modulo n). ... Group theory is that branch of mathematics concerned with the study of groups. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

• 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|. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... 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, injections, and surjections, and another which uses cardinal numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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 g, ord(g) = 1 if and only if g is the identity. If every element in G is the same as its inverse (i.e., g = g^-1), then ord(g) = 2 and consequently G is abelian since ab = (bb)ab(aa) = b(ba)(ba)a = ba. 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, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

The relation between the two concepts is the following: if we write

<a> = {a^k : k an integer}

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...

ord(a) = ord(<a>).

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... In mathematics, most commonly, Lagranges theorem 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. This can be shown using the concept of left cosets of H...

As an immediate consequence of the above, we see that the order of every element of a group divides the order of the group.

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(g) is some power of p for every g in G [2]. In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... The term composite can refer to several different things: A composite number is an integer greater than one that is not a prime number. ... This article is about the mathematical group. ...

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 integers which are not both zero is the largest integer that divides both numbers. ...

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, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

If G is a finite group of order n, and d is a divisor of n, then the number of elements in G of order d is a multiple of φ(d). In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less or equal than n and coprime to n. ...

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. 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. ...

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 G — if... 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 certain non-trivial proper subgroups of G.

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. ...