Group Theory

Group Theory

• inverse mapping
  $f{f}^{-1}=f^{-1}f=e$

• multiply
  $C=A\times B=\left\{(a_i,b_j),\mid a_i\in A,b_j\in B\right\}$

• binary opration
  $R:A\times A\to A,orR:(a,b)\to c=R(a,b)$
  $c=a\cdot b$

• homomorphism
  $f(x_i\cdot x_j)=f(x_i)\times f(x_j)$
  $\phi (g{g}^{\prime })=\phi (g)\phi (g^{\prime })$

• kernels
  The kernel of a homomorphism $\alpha :G\to G^{\prime }$
  $ker(\alpha )=\{g\in G|\alpha (g)=e\}$
  $G/ker(\alpha )\cong Imker(\alpha )$

• isomorphism
  $f(x_i\cdot x_j)\leftrightarrow f(x_i)\times f(x_j)$

• conjugate
  $g^{\prime }=hg{h}^{-1}$

• normal subgroups
  $gH{g}^{-1}=H\to H⊴G$

• quotient groups
  $H⊴G\to G=g_{\alpha }H\cup g_{\beta }H\cup g_{\gamma }H\cdots \to G/H=\{gH:g\in G\}$

• action of group
  left action G on X: $ G \times X \rightarrow X$
  written: $(g,x)\mapsto g\cdot x$
  another right action G on X: $X\times G\to X$
  written: $(g,x)\mapsto x\cdot g^{-1}$

• adjoint action
  $A{d}_g:G\to G$
  $A{d}_g{g}^{\prime }=g{g}^{\prime }{g}^{-1}$

• automorphism
  $\phi :G\to G$

• Set
  $A_1\cap A_2\cap A_3\cdots ={\bigcap }_{i=1}^n{A}_i$

• general linear group
  Let k be a field and choose $n\in N$ Then $G=GL(n,k)=G{L}_n(k)$ is defined to be the set of all invertible n × n matrices with entries in k called the general linear group degree n.
  For a finite-dimensional
  F-vector space V , the F-linear automorphisms of V form a group $GL(V) $ called the general linear group of V

• The quintic equation is unsolvable

$0=a_o+a_1{x}_1+a_2{x}_2^2+a_3{x}_3^3...$
$0=(x-x_1)(x-x_2)(x-x_3)...$
set: $M=\{1,2,...,n|n\ge 5\}$
symmetry group: $S_n=\{(1),(12),(13),(23),(123),(132),...\}$
normal subgroup: $gN=\{g,ga\}=Ng=\{g,ag\}\Rightarrow ag=ga$
$$\begin{gathered} H=gH{g}^{-1} \\ H=g^{-1}Hg \\ gH=g{g}^{-1}Hg=Hg \end{gathered}$$

$\{(1)\}⊴H_1⊴H_2⊴H_3...⊴H_n=G$
$K/F$
$F=F_0\subset F_1\subset F_2\subset ...\subset F_n=K$
domain isomorphism(Galois Group): $Gal(K/F)$
$\varphi (a)=a,\forall a\in F,\varphi \in Gal(K/F)$
$Gal(K/F)⊴S_n$

$\alpha \in \{a_1,a_2,a_3,...,a_m\},\varphi \in Gal(K/F)$
$0=\varphi (0)=\varphi (a_n{\alpha }^n+a_{n-1}{\alpha }^{n-1}+...+a_1\alpha +a_0)=\varphi (f(\alpha ))=f(\varphi (\alpha ))=a_n\varphi (\alpha {)}^n+a_{n-1}\varphi (\alpha {)}^{n-1}+...+a_1\varphi (\alpha )+a_0$

Define: $a(i)=j，1\le i，j\le n$
$g\in S_n,g(i)=i,g(j)=k\ne j$
$ag(i)=a(g(i))=a(i)=j,ga(i)=g(a(i))=g(j)=k\Rightarrow$ $ag(i)\ne ga(i)$

normalization constant:
$(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5)$
$x_1,x_2,x_3,x_4,x_5$
$c^m=Q \
(c^m / Q -1) = 0 \
c^m = 1 \
c = e^{2\pi i/mb}, b \in [1,2,3,4,\cdots m]
$
$f:c\to c^a,g:c\to c^b|f\ast g=g\ast f=c^{(a+b)}$

$f(x)=x^n+t_{n-1}{x}^{n-1}+t_{n-2}{x}^{n-2}+t_{n-3}{x}^{n-3}...t_{1}x+t_0$
$Q(t_0,t_1,t_2...t_{n-1})$
$K=Q(t_0,t_1,t_2,...,t_{n-1})(x_1,x_2,x_3...x_n)$

$Gal(K/O)⊴Gal(K/F)$
$Gal(O/F)\simeq Gal(K/F)/Gal(K/O)$

References

[1] S4 S5的子群

[2] 二阶子群

[3] @misc{milneGT,author={Milne, James S.},title={Group Theory (v3.16)},year={2020},note={Available at www.jmilne.org/math/},pages={137}}