In mathematics, the order of a finite group is the number of its elements. If a group 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, the order of an element of a group, is thus the smallest positive integer such that, where denotes the identity element 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
\operatorname{ord}(\langlea\rangle),
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 of . In particular, the order of any element is a divisor of .
The symmetric group 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 . 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 .
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.
ab=(ab)-1=b-1a-1=ba
2+2+2=6\equiv0\pmod{6}
The relationship between the two concepts of order is the following: if we write
\langlea\rangle=\{ak\colonk\inZ\}
\operatorname{ord}(a)=\operatorname{ord}(\langlea\rangle).
For any integer k, we have
ak = 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
ord(G) / ord(H) = [''G'' : ''H''], where [''G'' : ''H''] is called the index 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(S3) = 6, the possible orders of the elements are 1, 2, 3 or 6.
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]
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(ak) = ord(a) / gcd(ord(a), k)[3] for every integer k. In particular, a and its inverse a−1 have the same order.
In any group,
\operatorname{ord}(ab)=\operatorname{ord}(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(Z)
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.
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 homomorphisms or no injective homomorphisms, between two explicitly 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.
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:
|G|=|Z(G)|+\sumidi