Proposition 43 • Statement ○ If G is a group, H≤G, and [G:H]=2, then H⊴G • Proof ○ If g∈H, then gH=H=Hg ○ If g∉H, then gH=G∖H=Hg ○ Therefore gH=Hg,∀g∈G ○ So H⊴G • Corollary ○ Let p be the smallest prime dividing |G| ○ If [G:H]=p, then H⊴G ○ (Not ready to prove this yet) Proposition 44 • Statement ○ If (a_1…a_t ),(a_1′…a_t′) are t-cycles in S_n ○ Then ∃σ∈S_n s.t. σ(a_1…a_t ) σ^(−1)=(a_1′…a_t′) • Proof ○ Choose σ∈S_n s.t. σ(a_i )=a_i^′, ∀i∈{1,…,t} ○ By HW 7 #1, σ(a_1…a_t ) σ^(−1)=(σ(a_1 )…σ(a_t ))=(a_1′…a_t′) Theorem 45 • Statement ○ A_4 have no subgroup of order 6 • Proof ○ By way of contradiction, suppose H≤G, and |H|=6 ○ Then [A_4:H]=2 and thus H⊴A_4 ○ A_4 contains 8 3-cycles, so H contains some 3-cycle α ○ Write α=(a b c), then § (a b d)(a b c) (a b d)^(−1)=(b d c)∈H § (b c d)(a b c) (b c d)^(−1)=(a c d)∈H § (b d c)(a b c) (b d c)^(−1)=(a d b)∈H ○ So far, we have (1), (a b c),(b d c),(a c d),(a d b)∈H ○ Also, since H is closed under inverses, (a c b),(b c d)∈H ○ Thus, |H|≥7, which makes a contradiction ○ Therefore A_4 have no subgroup of order 6 Group Action • Definition ○ An action of G on X is a function G×X→X, (g,x)↦gx where ○ 1_G x=x, ∀x∈X ○ g(h�)=(ghx,∀g,h∈G,x∈X • Examples Set Group Action Rn GL_n (R (A,v)↦Av {1,…,n} S_n (σ,i)↦σ(i) Group G Group G (g,h↦gh Group G Group G (g,h↦ghg^(−1) Set of cosets of H≤G Group G (g,g^′ H)↦gg^′ H Set of all subgroups of group G Group G (g,H)↦gHg^(−1) • Note ○ If H≤G, and g∈G, then gHg(−1)={gh�(−1)│hH}≤G ○ gHg^(−1)≠∅, since g1g(−1)=1∈gHg(−1) ○ If ghg^(−1),gh′ g(−1)∈gHg(−1), then ○ ghg^(−1) (gh′ g^(−1) )(−1)=ghg(−1) g(h′ )^(−1) g^(−1)=gh(h′ )(−1)∈gHg(−1) Orbit and Stabilizer • Suppose a group G acts on a set X. Let x∈X • The orbit of x, denoted orb(x), is {gx│g∈G}⊆X • The stabilizer of x, denoted stab(x), is {g∈G│gx=x}⊆G Proposition 46 • Statement ○ If G acts on X, and x∈X, then stab(x)≤G • Proof ○ stab(x) ≠∅, because 1x=x ○ Let g,h∈stab(x) ○ (ghx=g(h�)=gx=x⇒gh∈stab(x) ○ x=1⋅x=(g^(−1) g)x=g^(−1) (gx)=g^(−1) x⇒g^(−1)∈stab(x) Centralizer • Let G be a group, and let G act on itself by conjugation • If h∈G, then stab(h={g∈G│gh�^(−1)=h={g∈G│ghh�} • This set is called the centralizer of h, denoted as C_G (h Center • Intersections of subgroups are subgroup • Thus if G acts on a set X, ⋃8_(x∈X)▒stab(x) ≤G • In the example above, ⋃8_(hG)▒〖C_G (h 〗=Z(G) is called the center of G Normalizer • Let G be a group, and let X be the set of subgroups of G • G acts on X by g⋅H=gHg^(−1) • If H≤G, then • stab(H)={g∈G│gHg^(−1)=H}={g∈G│gH=Hg} • This set is called the normalizer of H in G, denoted N_G (H) • N_G (H)=G⟺H⊴G
