Definition:Order of Group Element

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Definition

Let $G$ be a group whose identity is $e_G$.

Let $x \in G$ be an element of $G$.


Definition 1

The order of $x$ (in $G$), denoted $\order x$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.


Definition 2

The order of $x$ (in $G$), denoted $\order x$, is the order of the group generated by $x$:

$\order x := \order {\gen x}$


Definition 3

The order of $x$ (in $G$), denoted $\left\vert{x}\right\vert$, is the largest $k \in \Z_{\gt 0}$ such that:

$\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$


Infinite Order

$x$ is of infinite order, or has infinite order if and only if there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:

$\order x = \infty$


Finite Order

$x$ is of finite order, or has finite order if and only if there exists $k \in \Z_{> 0}$ such that $x^k = e_G$.


Examples

Order of $2$ in $\struct {\R_{\ne 0}, \times}$

Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

The order of $2$ in $\struct {\R_{\ne 0}, \times}$ is infinite.


Order of $i$ in $\struct {\C_{\ne 0}, \times}$

Consider the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.

The order of $i$ in $\struct {\C_{\ne 0}, \times}$ is $4$.


Order of $\begin{bmatrix} 1 & 1 \cr 0 & 1 \end{bmatrix}$ in General Linear Group

Consider the general linear group $\GL 2$.

Let $\mathbf A := \begin{bmatrix} 1 & 1 \cr 0 & 1 \end{bmatrix} \in \GL 2$

The order of $\mathbf A$ in $\GL 2$ is infinite.


Rotation Through the $n$th Part of a Full Angle

Let $G$ denote the group of isometries in the plane under composition of mappings.

Let $r$ be the rotation of the plane about a given point $O$ through an angle $\dfrac {2 \pi} n$, for some $n \in \Z_{> 0}$.

Then $r$ is the generator of a subgroup $\gen r$ of $G$ which is of order $n$.


Possible Orders of $x$ when $x^2 = x^{12}$

Let $G$ be a group.

Let $x \in G \setminus \set e$ be such that $x^2 = x^{12}$.

Then the possible orders of $x$ are $2$, $5$ and $10$.


Also known as

Some sources refer to the order of an element of a group as its period.


Also denoted as

The order of an element $x$ in a group is sometimes seen as $\map o x$.

Some sources render it as $\map {\operatorname {Ord} } x$.


Also see

  • Results about order of group elements can be found here.