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Algebra II

2021/2/22 - Recap and Roadmap for this semester

Recap (Some missing)

  • Group
    • cyclic
    • abelien
      • product of cyclic
    • non-abelian
      • Sn/An/Dn/Q
  • Ring
    • Integral Domain (No zero divisor)
      • Field
    • Field Extension (with multiplicative inverse)
      • finite/algebraic/transcendental
      • introduction to Galois(\(F_{P^k}\))

Roadmap

Hilight: CRT, RSA,UFD, GF, FTA

  • Group
    • group action
      • Cauchy Theorem
      • Burnside's Lemma
    • product of two subgroups
    • existence of normal subgroup
    • Sylow Theorems
      • existence of subgroup of certain size
      • when it will be normal
      • proof corollary: \(A_5\) is the smallest simple group
        • Note: simple group := group w/o normal subgroup
  • Ring
    • \(Z/nZ\)
      • Chinese Remainder Theorem
        • RSA
    • non-communative ring
      • quaternion algebra and 3D rotation
    • 3 Integral Domain
      • PID
      • UFD
        • Fermat's Last Theorem, case n=3
      • ED
  • Field
    • Galois Theory
      • Finite
      • Infinite
      • i.e. study field extension with group
      • applications
        • Foundamental Theorem of Algebra
        • Solving equation by Radical
        • Three Greek Geometric Problems

2021/2/25 - Group Action, Stabilizer and Orbit

Group Action

  • Definition
    • \(*:G\times X \to X\), \(G\) is a group, \(X\) is a set
    • \(*(g, x) \text{ write as } gx\)
    • satisfy
      • \(*(e,x)=x \forall x\)
      • \(*(g_2, *(g_1, x)) = *(g_2g_1,x)\)
    • another point of view
      • \(\rho:G\to S_X\) s.t. \(*(g,x)=\rho(g)(x)\)
      • group homomorphism \(G \to S_X\)
  • Remarks
    • think of its kernel!
  • Examples
    • Group Action of \(G\) on \(X=G\)
      1. left multiplication
        • \(\rho:G\to S_G, \rho(g)(x)=gx\)
      2. conjugate
        • \(\rho:G\to Aut(G), \rho(g)(x)=gxg^{-1}\)
    • Group Action of \(G\) on \(X=G/H\)
      • left multiplication
        • \(ker_{\rho}=\bigcap_{x\in G}xHx^{-1}=\text{the largest normal subgroup contained in H}\)
  • Additional Property
    • faithful
      • \(e \text{ is the only element in G that } ex = x \forall x \in X\)
    • transitive
      • \(\forall x_1, x_2, \exists g\in G \ni x_1=gx_2\)

Stabilizer and Orbit

  • Definition
    • \(G_x = \{g \in G :~gx=x \text{ for some }g\}\)
    • \(G_X = \{x \in X :~gx_0=x \text{ for some }g\}\) for some \(x_0\)
      • defined by eq. class induced partition
  • Thm.
    • \(|G_X| = [G:G_x]\)
      • proof by isomorphism
  • Thm. - the theorem of fixed subset of p-groups
    • \(|X| \equiv |X^G| \mod p\) when \(G\) is p-group
      • \(X^G = \bigcap_g X^g\), \(X^g=\{x\in X: gx=x\}\)

Applications of "theorem of fixed subset of p-groups"

  • Lemma - group of order \(p\) is cyclic (\(Z_p\))
    • pf: Lagrange
  • Thm. - p-group has non-trivial \(Z(G)\)
  • Class Equation
  • Thm. - group of order \(p^2\) is abelian
    • PBC, suppose not, \(Z(G), G/Z(G) \cong Z_p\)
  • Thm. - Cauchy thm.
    • suppose \(p| |G|\), \(G\) has an element of order \(p\)
    • pf sketch
      • \(X = \{(g_1,g_2,...,g_p) : g_1...g_p=e\}\)
      • \(P = <(1,2,...,p)>~\subset S_p\)
      • \(X^P\) has non-trivial element, but it can only be in the form of \((g,g,g...,g)\)

Burnsides, application and \(H\wedge N\)

Last update: 2023-09-15