Affine Group of One Dimension is Group

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Theorem

The $1$-dimensional affine group on $\R$ $\operatorname{Af}_1 \left({\R}\right)$ is a group.


Proof 1

Taking the group axioms in turn:

G0: Closure

Let $f_{ab}, f_{cd} \in \operatorname{Af}_1 \left({\R}\right)$.

Then:

\(\displaystyle f_{ab} \circ f_{cd} \left({x}\right)\) \(=\) \(\displaystyle a \left({c x + d}\right) + b\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle a c x + a d + b\) $\quad$ $\quad$

By the field axioms, $a c \in \R_{\ne 0}$ and $a d + b \in \R$.

Thus $f_{ab} \circ f_{cd} \in \operatorname{Af}_1 \left({\R}\right)$ and so $\operatorname{Af}_1 \left({\R}\right)$ is closed.

$\Box$


G1: Associativity

From Composition of Mappings is Associative, it follows directly that $\circ$ is associative on $\operatorname{Af}_1 \left({\R}\right)$.

$\Box$


G2: Identity

By Identity of Affine Group of One Dimension, $\operatorname{Af}_1 \left({\R}\right)$ has $f_{1, 0}$ as an identity element.

$\Box$


G3: Inverses

By Inverse in Affine Group of One Dimension, every element $f_{a b}$ of $\operatorname{Af}_1 \left({\R}\right)$ has an inverse $f_{c d}$ where $c = \dfrac 1 a$ and $d = \dfrac {-b} a$.

$\Box$


All the group axioms are thus seen to be fulfilled, and so $\operatorname{Af}_1 \left({\R}\right)$ is a group.

$\blacksquare$


Proof 2

It follows from Affine Group of One Dimension as Semidirect Product and Semidirect Product of Groups is Group.

$\blacksquare$


Sources