Subgroup/Examples/Matrices (1 a, 0 1) in General Linear Group

Example of Subgroup

Let $\GL 2$ denote the general linear group of order $2$.

Let $H$ be the set of square matrices of the form $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}$ for $a \in \R$.

Then $\struct {H, \times}$ is a subgroup of $\GL 2$, where $\times$ is used to denote (conventional) matrix multiplication.

Proof

We have that the unit matrix $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}$ is in $H$, so $H \ne \O$.

Then we have:

$\forall a, b \in \R: \begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & b \cr 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & a + b \cr 0 & 1 \end{bmatrix} \in H$

Then:

$\forall a \in \R: \begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & -a \cr 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 + b \cr 0 & 1 \end{bmatrix} \in H$

Thus $\begin{bmatrix} 1 & -a \cr 0 & 1 \end{bmatrix}$ is the inverse of $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}$ and so $\begin{bmatrix} 1 & a \cr 0 & 1 \end{bmatrix}^{-1} \in H$.

Hence the result from the Two-Step Subgroup Test.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Exercise $1 \ \text{(c)}$