Introduction to Group Structures

1. Group Actions (1.1) A group action of a group G acting on a set S is a map from G × S to S (written as g · s or g(s) for g ∈ G and s ∈ S) such that g1 · (g2 · s) = g1g2 · s and eG · s = s. If a group G is acting on a set S, we call S a G-set. Suppose S is a G-set. Define a relation s1 ∼ s2 ⇐⇒ s1 = g(s2) for some g ∈ G, then ∼ is an equivalence relation. (The equivalence classes are called orbits of S under G. If an orbit contains s ∈ S, then the orbit is called the orbit of s. The orbit of s := sG = {g(s)|g ∈ G}.) Proof: Check definition of equivalence relation.