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Concept
Order (group theory)
Summary
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 .
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