M. A. Armstrong

The Euclidean Group

The isometries of the plane form a group under composition of functions, the so called Euclidean group E2. A function g: R2 ia> R2 belongs to E2, provided it preserves distance; that is to say

Finite Rotation Groups

||g\left( x \right) - g\left( y \right)|| = ||x - y||

Subgroups and Generators

(1) for every pair of points x, y in ℝ2. If g, h ∈ E2, we have

Row and Column Operations

||g\left( {h\left( x \right)} \right) - g\left( {h\left( y \right)} \right)|| = ||h\left( x \right) - h\left( y \right)||

Actions, Orbits, and Stabilizers

(2) because g is an isometry