Given an element of group (which henceforth we abbreviate simply as ), the order of is the number of times we must add to itself to obtain the identity element .
Equivalently, it is the order of the group generated by : that is, the order of under the inherited group operation .
Conventionally, the identity element itself has order .
In the cyclic group , the order of the generator is . If we view as being the integers modulo under addition, then the element has order ; the elements and have order ; the elements and have order ; and the element has order .
In the group of integers under addition, every element except has infinite order. itself has order , being the identity.