64 Chapter 1. Groups I not mention: permutations are functions. We shall see that there are interesting consequences when this feature is restored. Definition. A group G acts on a set X if there is a function G×X → X, denoted by (g, x) → gx, such that (i) (gh)x = g(hx) for all g, h ∈ G and x ∈ X, (ii) 1x = x for all x ∈ X, where 1 is the identity in G. If G acts on X, we also call X a G-set. If a group G acts on a set X, then fixing the first variable, say g, gives a function αg : X → X, namely, αg : x → gx. This function is a permutation of X, for its inverse is αg−1 : αgαg−1 = α1 = 1X = αg−1 αg. It is easy to see that α: G → SX , defined by α: g → αg, is a homomorphism. Conversely, given any homomorphism ϕ: G → SX , define gx = ϕ(g)(x). Thus, an action of a group G on a set X is merely another way of viewing a homomorphism G → SX . Cayley’s Theorem says that a group G acts on itself by (left) translation, and its generalization, Theorem 1.96, shows that G also acts on the family of (left) cosets of a subgroup H by (left) translation. Example 1.99. We show that G acts on itself by conjugation. For each g ∈ G, define αg : G → G to be conjugation αg(x) = gxg−1. To verify axiom (i), note that for each x ∈ G, (αgαh)(x) = αg(αh(x)) = αg(hxh−1) = g(hxh−1)g−1 = (gh)x(gh)−1 = αgh(x). Therefore, αgαh = αgh. To prove axiom (ii), note that α1(x) = 1x1−1 = x for each x ∈ G, and so α1 = 1G. The following definitions are fundamental. Definition. If G acts on X and x ∈ X, then the orbit of x, denoted by O(x), is the subset O(x) = {gx : g ∈ G} ⊆ X the stabilizer of x, denoted by Gx, is the subgroup Gx = {g ∈ G : gx = x} ⊆ G. The orbit space, denoted by X/G, is the set of all the orbits. If G acts on a set X, define a relation on X by x ≡ y in case there exists g ∈ G with y = gx. It is easy to see that this is an equivalence relation whose equivalence classes are the orbits. The orbit space is the family of equivalence classes. Let us find some orbits and stabilizers.

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