Groups

Abstract

Let ρ be a rotation of 90° about the origin O in the plane. (Remember that convention dictates that ρ is then a counterclockwise rotation of 90°.) So, ρ ² denotes the rotation of 180° about O, and ρ ³ denotes the rotation of 270° about O, that is, p³ denotes 3 successive rotations of 90° about O. Let σ be the reflection in the X-axis and ı = σ ². Then ı is the identity mapping on the plane, sending each point of the plane to itself, and σ maps the point (x, y) to the point (x, −y). We write σ((x,y)) = (x, −y). Mappings, like these, that fix distance are called isometries and are multiplied under composition. This means, for example, that σρ, which is usually read “sigma rho,” can be more completely read “sigma following rho” and is defined by the formula

$$\sigma \rho (P) = \sigma (\rho (P))$$

for any point P.