Order of Group Element/Examples/Matrix (1 1, 0 1) in General Linear Group

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 $1$