Order (group theory)

In group theory, a branch of mathematics, the order of a group is its cardinality, that is, the number of elements in its set. If the group is seen multiplicatively, the order of an element "a" of a group, sometimes also called the period length or period of "a", is the smallest positive integer "m" such that , 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, "a" is said to have infinite order. 
The order of a group "G" is denoted by ord("G") or , and the order of an element "a" is denoted by ord("a") or . The order of an element "a" is equal to the order of its cyclic subgroup ⟨"a"⟩ = {"ak" for "k" an integer}, the subgroup generated by "a". Thus, = .
Lagrange's theorem states that for any subgroup "H" of "G", the order of the subgroup divides the order of the group: |"H"| is a divisor of |G|. In particular, the order |"a"| of any element is a divisor of |"G"|.
Example.
The symmetric group S3 has the following multiplication table.
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 the factorization 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"2 = "e"), then ord("a") = 2; this implies "G" is abelian since formula_1. The converse is not true; for example, the (additive) cyclic group Z6 of integers modulo 6 is abelian, but the number 2 has order 3:
The relationship between the two concepts of order is the following: if we write
for the subgroup generated by "a", then
For any integer "k", we have
In general, the order of any subgroup of "G" divides the order of "G". More precisely: if "H" is a subgroup of "G", then
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. 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":
for every integer "k". In particular, "a" and its inverse "a"−1 have the same order.
In any group, 
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 formula_6. 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 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".
Counting 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 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.
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 except 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.
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:
where the "di" 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.