代数讨论班1-群作用

Presentation For Seminar on Algebra

Yiping Lu 2016/2/23

Finite Group Theory

Introduction

所有显示问题都是张小帅的锅

post-classification period:

W.Burnside(the beginning of 20th centery):find and classify the finite simple gruops,the simple group classification now appears to be complete.

the classification of finite simple groups:

  • $Z_p$($p$ is a prime)

  • $A_n(n\ge5)$

  • Lie simple group(16)

  • 散在单群

other aspects of finite group theory:

  • permutation groups
  • p-groups
  • presentation theory

Group Action

Def:$G$is a group,and$\Omega$is a set,define a function:$G\times\Omega\rightarrow\Omega$and :

  • $\alpha,1=\alpha(\alpha\in\Omega)$
  • $(\alpha,g),h=\alpha,gh(\alpha\in\Omega,g,h\in G)$

kernel:we can build a mapping between$G$and$Sym(\Omega)$ as follow: $$\sigma:G\rightarrow Sym(\Omega):\sigma(g)(a)=ga (g\in G,a\in\Omega)$$

*An action is an iso from $G$ to $Sym(\Omega)$ *

$ker(\sigma)$is the kernel of the group action

faithful:kernel=1

Orbit: $Orb(x)$ is a Equivalence class of $a\sim b(\leftrightarrow g(a)=b)$

$$|S|=\sum_{i=1}^{t}|Orb(s_i)|$$

stabilizer:for x in$\Omega$,the stabilizer of $x$ is defined as below:

$$Stab(s):={g\in G|g(s)=s}$$

at the same time the following statement is very simple but useful:

$$kernel=\cap_{x\in\Omega}Stab(x)$$

if$g(s)=s'$ then
$$Stab(s')=gStab(s)g^{-1}$$ also we can have:

The Fundamental Counting Principle $$|Orb(s)|=|G:Stab(s)|$$ (build a bijection between${Hx|x\in G}$and$orb(s)$) we can also use $\mathscr{O}$ to present orbit

lemma:

homo $\phi:X\rightarrow Y$,
$x\in X$then $ Stab_G(x)\le Stab_G({\phi(x)}) $ ,if $\phi$ is an iso then

$$Stab_G(x)\simeq Stab_G({\phi(x)})$$

proof: for every$g\in Stab_G(x)$,we have$\phi(x)=\phi(gx)=g\phi(x)$

Examples of Group Action:

\begin{longtable}{|c|c|c|c|c|}
\hline
\hline
Action&Sets & rules &Stablizer&kernel\\  
\hline
\hline
      \textbf{regular action}&$G$      & $x\times g=xg$ &1&faithful \\
\hline
      \textbf{conjugation action}&$G$     & $x\times g=g^{-1}xg:=x^g$   &$C_G(x)$&$Z(G)$\\
\hline
&subsets of $G$&$X\times g=X^g:=\{x^g|x\in X\}$&$N_G(x)$&$Z(G)$\\
\hline
\textbf{right multiplication}      &cosets of $H$       & $Hx\times g=Hxg$  & $H^x$ &$Core_G(H)$\\
\textbf{action on cosets}& & & &\\
\hline
\caption{Some example of Group Action}
\end{longtable}

Collary:Caley Thm$|G|=n$,then $G$ is iso to a subgroup of $S_n$

Normalizer:

$$N_G(S)={x\in G|xSx^{-1}=S}$$

Centralizer:

$$C_G(S)={x\in G|xa=ax,a\in S}$$

we have $C_G(S)\triangleleft N_G(S),N_G(S)\le G$

Core of H on G:

$$Core_G(H):=\cap_{x\in G}H^x$$

for every$N\triangleleft G,N\le H$,we have:$N=N^x\subset H^x$,so$N\subset Core_G(H)$,at the same time$Core_G(H)\triangleleft G$,so the Core is the biggest normal subgroup of $G$ in H.

conjugation action

Statement$x$ in group $G$ which is a finite group, and let $K$ be the conjugacy class of $G$ containing $x$.Then$|K|=|G:C_G(x)|$

Class equation of $G$: $|G|=|Z(G)|+\sum_{j=1}^s|G:C_G(y_j)|$

Corollary:$G$is a $p-$group,then$|Z(G)|\not=1$. Moreover,$|Z(G)|$is a $p-$group has at least $p$ elements

Corollary: for every$H\le G$,

$$\cup_{g\in G}gHg^{-1}\not=G$$

right multiplication action on cosets

a bijection $\delta:G/core_G(H)\rightarrow Sym(\Omega)$ defined as following:
$$\delta(gCore_G(H))(Ha)=Hag$$ the function is well defined and $kernel={Core_G(H)}$ which means faithful
Corollary:$G\le H$and$|G:H|=n$Then $H$ contains a normal subgroup $N$,$|G:N||n!$($N=Core_G(H)$) use this corollary we can simply have the example follows:

example:if$|G:H|$is the smallest prime divisor of$|G|$,then$H\triangleleft G$

Corollary:A simple group $G$ contains a subgroup of index $n\ge 1$,then$|G||n!$

example:$H,K\le G$,if$G/H\simeq G/K$then$H=gKg^{-1}$

proofFirst:$Stab(Hx)=x^{-1}Hx,Stab(Gx)=x^{-1}Gx$

lemma:homo$\phi:X\rightarrow Y$,$x\in X$then$Stab_G(x)\le Stab_G({\phi(x)})$,if$\phi$is an iso then $Stab_G(x)\simeq Stab_G({\phi(x)})$

(which is proved before)

at the same time$G/H\simeq G/K$,then$H$is also a stablizer of $G/K$,it is $H=gKg^{-1}$

reference

  • Finite Group Theory I.martin Isaacs

  • Abstract Algebra(I) Chunlai Zhao

  • Abstract Algebra Qinhai Zhang

  • Group Represent Theory(GTM162)

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