Order of a group element

Written by Patrick Stevens last updated

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 .

Examples

In the symmetric group , the order of an element is the least_common_multiple of its cycle type.

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.

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