Transformation Groups and Erlangen Program - Key Properties

The structure of transformation groups determines geometric properties. Understanding group actions reveals deep connections between algebra and geometry.

DefinitionOrbits and Stabilizers

For a group $G$ acting on space $X$ :

The orbit of $x \in X$ is $G \cdot x = \{g(x) : g \in G\}$
The stabilizer of $x$ is $G_x = \{g \in G : g(x) = x\}$

The orbit-stabilizer theorem: $|G| = |G \cdot x| \cdot |G_x|$ (for finite groups).

Homogeneous spaces arise when $G$ acts transitively. Every point "looks the same"—there's no distinguished origin. Euclidean space is homogeneous under translations; the sphere is homogeneous under rotations.

ExampleKlein's Models

Klein constructed hyperbolic geometry models using transformation groups:

Poincaré disk: $PSU(1,1)$ acts by Möbius transformations
Klein disk: $PO(2,1)$ acts by projective transformations
Hyperboloid: $SO^+(2,1)$ acts by Lorentz transformations

Different models correspond to different group representations, but all realize the same geometric structure.

DefinitionInvariants

An invariant of a group action is a function $f: X \to Y$ satisfying $f(g(x)) = f(x)$ for all $g \in G$ , $x \in X$ .

Geometric properties are invariants: distance is an invariant of isometries, cross-ratio is an invariant of projectivities.

Remark

Modern gauge theories in physics are geometries in Klein's sense: spacetime with symmetry groups (Lorentz, gauge). Noether's theorem connects symmetries (transformation groups) to conservation laws, exemplifying the power of the Erlangen viewpoint in theoretical physics.