Order (group theory)

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In group theory, the order of a group element is the least positive integer (if one exists) such that raising the element to that power gives the identity element of the group. If there is no such number, the element is said to be of infinite order.

The order of a group is just its cardinality as a set. The connexion between the two is that the order of an element is equal to the order of the cyclic group generated by that element.

The exponent of a group is the least positive integer (if one exists) such that raising any element of the group to that power gives the identity. The exponent can be evaluated as the least common multiple of the orders of the elements. For an Abelian group, there is always an element whose order is equal to the exponent.