Order (group Theory)

Order (group Theory)

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

• The order of a group is its cardinality, i.e., the number of elements in its set.
• The order, sometimes period, of an element a of a group is 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, a is said to have infinite order. All elements of finite groups have finite 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|.