Order of Group Element/Examples
Examples of Order of Group Element
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 $10$.
Non-Cyclic Group of Order $55$ has Order $5$ Element and Order $11$ Element
Let $G$ be a non-cyclic group whose order is $55$.
Then $G$ has: