# Affine Group of One Dimension is Group

## Theorem

Let $\map {\operatorname {Af}_1} \R$ be the $1$-dimensional affine group on $\R$.

Then $\map {\operatorname {Af}_1} \R$ is a group.

## Proof 1

Taking the group axioms in turn:

### Group Axiom $\text G 0$: Closure

Let :

$a, c \in \R_{\ne 0} \land b, d \in \R$

Let:

$f_{ab}, f_{cd} \in \map {\operatorname {Af}_1} \R$

Then:

 $\ds \map{\paren{f_{ab} \circ f_{cd} } } x$ $=$ $\ds \map{f_{ab} } {\map{f_{cd} } x}$ Definition of Composition of Mappings $\ds$ $=$ $\ds \map{f_{ab} } {c x + d}$ Definition of Affine Group of One Dimension $\ds$ $=$ $\ds a \paren {c x + d} + b$ Definition of Affine Group of One Dimension $\ds$ $=$ $\ds \paren{a c x + a d } + b$ Real Number Axiom $\R \text A1$: Associativity of Addition $\ds$ $=$ $\ds \paren{a c x} + \paren{a d + b }$ Real Number Axiom $\R \text A1$: Associativity of Addition $\ds$ $=$ $\ds \paren{a c } x + \paren{a d + b }$ Real Number Axiom $\R \text M1$: Associativity of Multiplication

By the field axioms:

$a c \in \R_{\ne 0} \land a d + b \in \R$

Thus $f_{ab} \circ f_{cd} \in \map {\operatorname {Af}_1} \R$ and so $\map {\operatorname {Af}_1} \R$ is closed.

$\Box$

### Group Axiom $\text G 1$: Associativity

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

$\Box$

### Group Axiom $\text G 2$: Existence of Identity Element

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

$\Box$

### Group Axiom $\text G 3$: Existence of Inverse Element

By Inverse in Affine Group of One Dimension, every element $f_{a b}$ of $\map {\operatorname {Af}_1} \R$ 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 $\map {\operatorname {Af}_1} \R$ is a group.

$\blacksquare$

## Proof 2

$\blacksquare$