Tour:Order of an element
This article adapts material from the main article: order of an element
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WHAT YOU NEED TO DO: Understand the definition of order of an element given below.
Definition
The order of an element in a group
is the smallest positive integer
for which
is the identity element.
Such a may not always exist (if it exists,
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
in any group
- In the group of integers modulo
, the element
has order
WHAT'S MORE: Some further facts about orders of elements, related to content we'll be seeing later in the tour.
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 in a finite group
divides the order of
(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.
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