Abstract Algebra Theory and Applications - Computer Science ...

210 CHAPTER 12 GROUP ACTIONSLemma 12.6 Let X be a G-set and suppose that x ∼ y. Then G x is isomorphicto G y . In particular, |G x | = |G y |.Proof. Let G act on X by (g, x) ↦→ g · x. Since x ∼ y, there exists a g ∈ Gsuch that g · x = y. Let a ∈ G x . Sincegag −1 · y = ga · g −1 y = ga · x = g · x = y,we can define a map φ : G x → G y by φ(a) = gag −1 .homomorphism sinceThe map φ is aφ(ab) = gabg −1 = gag −1 gbg −1 = φ(a)φ(a).Suppose that φ(a) = φ(b). Then gag −1 = gbg −1 or a = b; hence, the map isinjective. To show that φ is onto, let b be in G y ; then g −1 bg is in G x sinceg −1 bg · x = g −1 b · gx = g −1 b · y = g −1 · y = x;and φ(g −1 bg) = b.□Theorem 12.7 (Burnside) Let G be a finite group acting on a set X andlet k denote the number of orbits of X. Thenk = 1|G|∑|X g |.g∈GProof. We look at all the fixed points x of all the elements in g ∈ G; thatis, we look at all g’s and all x’s such that gx = x. If viewed in terms of fixedpoint sets, the number of all g’s fixing x’s is∑|X g |.g∈GHowever, if viewed in terms of the stabilizer subgroups, this number is∑|G x |;x∈Xhence, ∑ g∈G |X g| = ∑ x∈X |G x|. By Lemma 12.6,∑y∈O x|G y | = |O x | · |G x |.