Modern Algebra II Orthogonal Transformations and Rotations

Group Operations Definition. Let G be a group and S a set. An operation of G on S is a rule for combining elements g ∈ G and s ∈ S so that gs ∈ S, such that 1s = s for all s, and (gg’)s = g(g’s) for all g, g’, and s. With this operation, S is called a G-set. Definition. Let s ∈ S, with S a G-set. The orbit of s is the set Os = {s’ ∈ S | s’ = gs for some g ∈ G}. Proposition. S is a union of disjoint orbits. Definition. If S consists of a single orbit, G operates transitively on S. Definition. The stabilizer of s ∈ S is the subgroup Gs = {g ∈ G | gs = s}. Proposition. xs = ys ó xy ∈ Gs. Definition. Let H be a subgroup of a group G. The set of left cosets, aH, of G is called the coset space, and may be written G/H. G/H is a G-set, under the operation g(aH) =