Set of Isometries in Complex Plane under Composition forms Group
Theorem
Let $S$ be the set of all complex functions $f: \C \to \C$ which preserve distance when embedded in the complex plane.
That is:
- $\size {\map f a - \map f b} = \size {a - b}$
Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$.
Then $\struct {S, \circ}$ is a group.
Proof
From Complex Plane is Metric Space, $\C$ can be treated as a metric space.
Hence it is seen that a complex function $f: \C \to \C$ which preserves distance is in fact an isometry on $\C$.
Taking the group axioms in turn:
Group Axiom $\text G 0$: Closure
Let $f$ and $g$ be isometries on $\C$.
We have:
\(\ds \forall a, b \in \C: \, \) | \(\ds \size {\map {\paren {g \circ f} } a - \map {\paren {g \circ f} } b}\) | \(=\) | \(\ds \size {\map g {\map f a} - \map g {\map f b} }\) | Definition of Composition of Mappings | ||||||||||
\(\ds \) | \(=\) | \(\ds \size {\map g a - \map g b}\) | $f$ is an Isometry | |||||||||||
\(\ds \) | \(=\) | \(\ds \size {a - b}\) | $g$ is an Isometry |
Thus $g \circ f$ is an isometry, and so $\struct {S, \circ}$ is closed.
$\Box$
Group Axiom $\text G 1$: Associativity
By Composition of Mappings is Associative, $\circ$ is an associative operation.
$\Box$
Group Axiom $\text G 2$: Existence of Identity Element
The identity mapping is the identity element of $\struct {S, \circ}$.
$\Box$
Group Axiom $\text G 3$: Existence of Inverse Element
By definition, an isometry is a bijection.
For $f \in S$, we have that $f^{-1}$ is also an isometry.
Thus every element of $f$ has an inverse $f^{-1}$.
$\Box$
All the group axioms are thus seen to be fulfilled, and so $\struct {S, \circ}$ is a group.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{B iii}$