Math 541 - 3/21 Home / 2018 / March / 23 / Math 541 - 3/21 Math 541 - 3/21 Mar 23, 2018 Shawn Math 541 No comments yet Related Posts Math 541 January 30, 2018 Shawn Leave a Reply Cancel replyYour email address will not be published. This site uses Akismet to reduce spam. Learn how your comment data is processed. 5.4 Recursive Algorithms6.1 The Basics of Counting Search Search for: Home Page Tutorials Mathematics Math 240 - Discrete Math Math 375 - Linear Algebra Math 431 - Intro to Probability Math 514 - Numerical Analysis Math 521 - Analysis I Math 541 - Abstract Algebra Math 632 - Stochastic Processes Abstract Algebra @ 万门大学 Linear Algebra @ 万门大学 Category Theory Computer Sciences CS/ECE 252 - Intro to Computer Engr. CS/ECE 352 - Digital System Fund. Learn Haskell Projects 2048 Game HiMCM 2016 登峰杯 MCM Course Notes AP Macroeconomics AP Microeconomics AP Chemistry AP Statistics AP Physics C: E＆M AP Physics C: Mechanics CLEP Psychology WeChat Account TOP
![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](https://shawnzhong.com/wp-content/uploads/2018/04/img_5ac3f59a5b440.png)
