Order of a group

In group theory, a branch of mathematics, the term order is used in two unrelated senses:

This article is about the first sense of order.

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|.

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 1. Each of s, t, and w squares to e, so these group elements have order 2. Completing the enumeration, both u and v have order 3, for u² = v and u³ = vu = e, and v² = u and v³ = uv = e.

The order of a group and that of an element tend to speak about the structure of the group. Roughly speaking, the more complicated the factorization of the order the more complicated the group.

If the order of group G is 1, then the group is called a trivial group. Given an element a, ord(a) = 1 if and only if a is the identity. If every (non-identity) element in G is the same as its inverse (so that a² = e), then ord(a) = 2 and consequently G is abelian since ${\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba}$ by Elementary group theory. The converse of this statement is not true; for example, the (additive) cyclic group Z₆ of integers modulo 6 is abelian, but the number 2 has order 3:

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