Affine Group of One Dimension as Semidirect Product
Theorem
Let $\map {\operatorname{Af}_1} \R$ be the $1$-dimensional affine group on $\R$.
Let $\R^+$ be the additive group of real numbers.
Let $\R^\times$ be the multiplicative group of real numbers.
Let $\phi: \R^\times \to \Aut {\R^+}$ be defined as:
- $\forall b \in \R^\times: \map \phi b = \paren {a \mapsto a b}$
Let $\R^+ \rtimes_\phi \R^\times$ be the corresponding semidirect product.
Then:
- $\map {\operatorname {Af}_1} \R \cong \R^+ \rtimes_\phi \R^\times$
where $\cong$ denotes (group) isomorphism.
Proof
By definition, a (group) isomorphism is a (group) homomorphism which is a bijection.
Recall the definition of underlying set of $1$-dimensional affine group on $\R$:
- $S = \set {f_{a b}: x \mapsto a x + b : a \in \R_{\ne 0}, b \in \R}$
So the bijection $\psi: \map {\operatorname {Af}_1} \R \to \R^+ \rtimes_\phi \R^\times$ defined by $\map \psi {f_{a b} } = \paren {b, a}$ arises naturally.
It remains to show that $\psi$ is a (group) homomorphism:
Let $f_{a b}, f_{c d} \in \map {\operatorname {Af}_1} \R$.
Then:
| \(\ds \map {f_{a b} \circ f_{c d} } x\) | \(=\) | \(\ds \map {f_{a b} } {\map {f_{c d} } x}\) | Definition of Composition of Mappings | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map {f_{a b} } { c x + d }\) | Definition of Affine Group of One Dimension | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \paren {c x + d} + b\) | Definition of Affine Group of One Dimension | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren{ a \paren{c x } + a d } + b\) | Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \paren{c x } + \paren{a d + b }\) | 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds f_{\paren {a c} \paren {a d + b} }\) | Definition of Affine Group of One Dimension |
Let $\tuple {b, a}, \tuple {d, c} \in \R^+ \rtimes_\phi \R^\times$.
Then:
| \(\ds \tuple {b, a} \tuple {d, c}\) | \(=\) | \(\ds \paren {b + \map {\map \phi a} d, a c}\) | Definition of Semidirect Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {b + a d, a c}\) | Definition of $\phi$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a d + b, a c}\) | Real Addition is Commutative |
So:
- $\map \psi {f_{a b} } \, \map \psi {f_{c d} } = \map \psi {f_{a b} \circ f_{c d} }$
So the bijection $\psi$ is a (group) homomorphism, and thus a (group) isomorphism.
$\blacksquare$