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 a of a group, is thus the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a. If no such m exists, the order of a is infinite. The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|, instead of ord ( ⟨ a ⟩ ) , {\displaystyle \operatorname {ord} (\langle a\rangle ),} where the brackets denote the generated group. Lagrange's theorem states that for any subgroup H of a finite group G, the order of the subgroup divides the order of the group; that is, |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 ord(S3) = 6. By definition, the order of the identity, e, is one, since e 1 = e. Each of s, t, and w squares to e, so these group elements have order two: |s| = |t| = |w| = 2. Finally, u and v have order 3, since u3 = vu = e, and v3 = uv = e.
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 a2 = e), then ord(a) = 2; this implies G is abelian since a b = ( a b ) − 1 = b − 1 a − 1 = b a {\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba} . 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:
2 + 2 + 2 = 6 ≡ 0 ( mod 6 ) {\displaystyle 2+2+2=6\equiv 0{\pmod {6}}} . The relationship between the two concepts of order is the following: if we write
⟨ a ⟩ = { a k : k ∈ Z } {\displaystyle \langle a\rangle =\{a^{k}\colon k\in \mathbb {Z} \}}
for the subgroup generated by a, then
ord ( a ) = ord ( ⟨ a ⟩ ) . {\displaystyle \operatorname {ord} (a)=\operatorname {ord} (\langle a\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. 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:
ord(ak) = ord(a) / gcd(ord(a), k) for every integer k. In particular, a and its inverse a−1 have the same order. In any group,
ord ( a b ) = ord ( b a ) {\displaystyle \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 S y m ( Z ) {\displaystyle Sym(\mathbb {Z} )} . 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.
… excerpt ends here. Continue reading the full article.
