Section 1.7. Group Actions 67 Proposition 1.105 (N/C Lemma). (i) If H ⊆ G, then CG(H) NG(H) and there is an imbedding NG(H)/CG(H) → Aut(H). (ii) G/Z(G) ∼ = Inn(G), where Inn(G) is the subgroup of Aut(G) consisting of all the inner automorphisms. Proof. (i) If a ∈ G, denote conjugation g → aga−1 by γa. Define ϕ: NG(H) → Aut(H) by ϕ: a → γa|H. Note that ϕ is well-defined, for γa|H ∈ Aut(H) because a ∈ NG(H). It is routine to check that ϕ is a homomorphism. Now the following statements are equivalent: a ∈ ker ϕ γa|H = 1H aha−1 = h for all h ∈ H a ∈ CG(H). The First Isomorphism Theorem gives CG(H) NG(H) and NG(H)/CG(H) ∼ = im ϕ ⊆ Aut(H). (ii) In the special case H = G, we have NG(H) = G, CG(H) = Z(G), and im ϕ = Inn(G). • Remark. We claim that Inn(G) Aut(G). If ϕ ∈ Aut(G) and g ∈ G, then ϕγaϕ−1 : g → ϕ−1g → aϕ−1ga−1 → ϕ(a)gϕ(a−1). Thus, ϕγaϕ−1 = γϕ(a) ∈ Inn(G). Recall that an automorphism is called outer if it is not inner the outer automorphism group is defined by Out(G) = Aut(G)/Inn(G). Proposition 1.106. If G acts on a set X, then X is the disjoint union of the orbits. If X is finite, then |X| = i |O(xi)|, where one xi is chosen from each orbit. Proof. As we have mentioned earlier, the relation on X, given by x ≡ y if there exists g ∈ G with y = gx, is an equivalence relation whose equivalence classes are the orbits. Therefore, the orbits partition X. The count given in the second statement is correct: since the orbits are disjoint, no element in X is counted twice. • Here is the connection between orbits and stabilizers. Theorem 1.107. If G acts on a set X and x ∈ X, then |O(x)| = [G : Gx], the index of the stabilizer Gx in G. Proof. Let G/Gx denote the family of all the left cosets of Gx in G. We will exhibit a bijection ϕ: G/Gx → O(x), and this will give the result, since |G/Gx| = [G : Gx]. Define ϕ: gGx → gx. Now ϕ is well-defined: if gGx = hGx, then h = gf for some f ∈ Gx that is, fx = x hence, hx = gfx = gx. Now ϕ is an injection: if gx = ϕ(gGx) = ϕ(hGx) = hx, then h−1gx = x hence, h−1g ∈ Gx, and gGx = hGx.

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