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 $x$ 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 $n$

WHAT'S MORE: Some further facts about orders of elements, related to content we'll be seeing later in the tour.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.
PREVIOUS: Group of integers modulo n| UP: Introduction four (beginners)| NEXT: Cyclic group