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 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 , and the order of an element a is denoted by ord(a) or , instead of 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, is a divisor of . In particular, the order of any element is a divisor of . The symmetric group S3 has the following multiplication table. {| class="wikitable" |- ! • ! 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 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 .