Order (group theory)

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 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, 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 ${\displaystyle \operatorname {ord} (\langle a\rangle ),}$ 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|.

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.

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 ${\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba}$. 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:

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