Definition:Group Action of GL(2,Q) on Irrational Numbers

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Let $\GL {2, \Q}$ be the general linear group on the field of rational numbers.

Let $\R \setminus \Q$ be the set of irrational numbers.

The standard group action of $\GL {2, \Q}$ on $\R \setminus \Q$ is the group action:

$\GL {2, \Q} \times \paren {\R \setminus \Q} \to \R \setminus \Q$:
$\paren {\begin {pmatrix} a & b \\

c & d \end {pmatrix}, x} \mapsto \dfrac {a x + b} {c x + d}$

Also see