Let A be the set of all 2 by 2 invertible matrices whose determinant is equal to 1.
(5)
\begin{align} A = \lbrace \blacklozenge \; \vert \; \blacklozenge \in GL_2(\mathbb{R}) \; ,\; det(\blacklozenge) = 1 \rbrace \end{align}
Closure
(6)
\begin{align} Let \; \blacklozenge_1 \in A \;and \; \blacklozenge_2 \in A \end{align}
(7)
\begin{align} det(\blacklozenge_1\blacklozenge_2) \; = det(\blacklozenge_1)det(\blacklozenge_2) \; = \; (1)(1) \; = \; 1 \end{align}
Identity
(8)
\begin{align} det(I) \; = \; (1)(1) \; - \; (0)(0) \; = \; 1 \end{align}
Inverse
(9)
\begin{align} det(\blacklozenge_1^{-1}) \; = \; det(\blacklozenge_1)^{-1} \; = \; \frac{1}{det(\blacklozenge_1)} \; = \; \frac{1}{1} \; = \; 1 \end{align}
Cosets
(10)
\begin{align} Let \; \spadesuit_1, \; \spadesuit_2, \; ..., \; \spadesuit_n \; \in \; GL_2(\mathbb{R}) \; and \; A \; \leq \; GL_2(\mathbb{R}) \end{align}
(11)
\begin{align} \spadesuit_1 A \; = \; \lbrace \spadesuit_1\blacklozenge_1, \; \spadesuit_1\blacklozenge_2, \; ...,\; \spadesuit_1\blacklozenge_n\rbrace \end{align}
(12)
\begin{align} \spadesuit_2 A \; = \; \lbrace \spadesuit_2\blacklozenge_1, \; \spadesuit_2\blacklozenge_2, \; ...,\; \spadesuit_2\blacklozenge_n\rbrace \end{align}
(13)
\begin{align} \bullet \end{align}
(14)
\begin{align} \bullet \end{align}
(15)
\begin{align} \bullet \end{align}
(16)
\begin{align} \spadesuit_n A \; = \; \lbrace \spadesuit_n\blacklozenge_1, \; \spadesuit_n\blacklozenge_2, \; ...,\; \spadesuit_n\blacklozenge_n\rbrace \end{align}