Affine Group of One Dimension is Group/Proof 1
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
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.
$\blacksquare$