Order of Group Element/Examples/Matrix (1 1, 0 1) in General Linear Group
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Examples of Order of Group Element
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.
Proof
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then we see that:
- $\mathbf A^n = \begin{bmatrix} 1 & n \cr 0 & 1 \end{bmatrix}$
and so:
- $\forall n \in \Z_{>0}: \mathbf A^n \ne \begin{bmatrix} 1 & 0 \cr 0 & 1 \end{bmatrix}$
Hence the result.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Definition $3.9$: Remark $1$