What does it mean for a element of a group to have a order?

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

The order of a group is its cardinality, i. e., the number of elements in its set. Also, 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. The ordering relation of a partially or totally ordered group.

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 |.