4 I. GROUPS AND COUNTING PRINCIPLES Definition. Let S be a G-space with action r. Let s G S. Then {rx(s) | x G G} is called the orfo£ OJ of s. If OJ = 5, we say that the action is transitive. A transitive G-space is sometimes called a homogeneous space for G. Since rx(s) = t if and only if r x -i (£) = s, it is easy to see that "t lies in the orbit of 5" is an equivalence relation, so Proposition 1.2.2. Any G-space is a disjoint union of its orbits. One way of looking at Example 2 is to think of G as an if-space with the action T h \9) ~ 9^" The left cosets of H are precisely the orbits under this action. Example 1, revisited. The orbits under the action rx(y) = xyx"1, that is, {xyx~l I x G G, y fixed}, are called conjugacy classes. We denote them as 1 Gi,... , C[. By the proposition, d C\ Cj = 0 and U d = G. They will play an important role in the third chapter. Example 2, revisited. Since rx(H) = xH, the orbit of H in G/H is all of G/H. Thus, G/H is a transitive G-space. We are heading toward the proof that every transitive G-space is a G/H. Definition. Let S be a G-space and let s G S. The isotropy subgroup Is of s is {x G G I TX(S) = s}. Notice that in Example 2, rx{H) = H implies x = xe G xH, so rx(H) = H if and only if a: G //" that is, Ijj = iJ. Notice in general that rx(ry(s)) = ry(s) if and only if ry-ixy(s) = s that is, lry(s) =yisVl- Definition. Let (S,T), (5, f) be two G-spaces. We say that S and S are iso- morphic if and only if there exists a tp : S — 5, a bijection so that 7^(^(5)) = ^ ( T ^ S ) ) all 5 G 5, X G G. Theorem 1.2.3 (Fundamental theorem of G-spaces). Let (5, r) be a tran- sitive G-space. Let Is be the isotropy group of some s G S. Then (5, r) is isomorphic to G/Is. In particular, #(S) = #(G)/#(/ f l ). Proo/. Fix s. Given ^ G 5, define Qs/ C G as Qs ={x G G | rx(5) = s/}. Fix x G Qs'- Then y G 3s' if and only if x~xy G Is, that is, if and only if y G x/ s . Thus, Qs = x/ s and the Qs/ are precisely elements of G/Is. Let V?(5') = Q8' e G/Is. Then V(7"x(s')) =xQs =x(p{sf), so (^ is the required isomorphism. •

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