Order (group theory)

In group theory, a branch of mathematics, the term order is used in three different senses: In group theory, a branch of mathematics, the term order is used in three different senses: This article is about the first two senses of order. They are closely related: 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. 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|. Thus, |a| = |⟨a⟩|. 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 ord(S3) = 6. By definition, the order of the identity, e, is one, since e1 = 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 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: