Order of an element

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WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with order of a group

Definition

The order of an element x in a group G is the smallest positive integer n for which x^n is the identity element.

Such a n may not always exist (if it exists, x is said to be of finite order, or is termed a torsion element). It does exist when the group is finite.

Examples

  • The identity element has order 1 in any group
  • In the group of integers modulo n, the element 1 has order n

Facts

For an element of finite order, the order of the element equals the order of the cyclic subgroup generated by the element. Thus, by Lagrange's theorem, the order of an element x in a finite group G divides the order of G (where order here means the total cardinality of the group).

The exponent of a group is defined as the least common multiple of the orders of all elements of the group. For a finite group, the exponent always exists, and is a divisor of the order of the group (though it may, in general, be smaller). There may or may not exist an element in the group whose order equals the exponent of the group.

For an infinite group, not every element necessarily has finite order. A group where every element has finite order is termed a periodic group. Even for a periodic group, the exponent may be infinite because there may not be a common bound on the orders of all elements. A group with bounded exponent is a group whose exponent is finite, the condition of having bounded exponent is stronger than the condition of being periodic.