交换群结构定理
G ≅ Z / d 1 Z × Z / d 2 Z × ⋯ × Z / d n Z , d 1 ∣ d 2 , ⋯ , d n − 1 ∣ d n G\cong \mathbb{Z}/d_1\mathbb{Z}\times \mathbb{Z}/d_2\mathbb{Z}\times\cdots\times \mathbb{Z}/d_n\mathbb{Z}, \quad d_1|d_2,\cdots,d_{n-1}|d_n G≅Z/d1Z×Z/d2Z×⋯×Z/dnZ,d1∣d2,⋯,dn−1∣dn
群在集合上的作用
集合 Ω \Omega Ω, 群 G G G, ρ \rho ρ是群同态, ρ ( g ) \rho(g) ρ(g)是双射.
ρ : G ⟶ S Ω g ⟼ ρ ( g ) \begin{aligned} \rho: G&\longrightarrow S_\Omega\\ g&\longmapsto \rho(g) \end{aligned} ρ:Gg⟶SΩ⟼ρ(g)
其中
ρ ( g ) : Ω ⟶ Ω ω ⟼ ρ ( g ) ( ω ) ≜ g ( ω ) \begin{aligned} \rho (g): \Omega&\longrightarrow \Omega\\ \omega&\longmapsto \rho(g)(\omega)\triangleq g(\omega) \end{aligned} ρ(g):Ωω⟶Ω⟼ρ(g)(ω)≜g(ω)
ρ \rho ρ保持运算, 即
ρ ( g h ) = ρ ( g ) ⋅ ρ ( h ) , \rho(gh)=\rho(g)\cdot \rho(h), ρ(gh)=ρ(g)⋅ρ(h),
⟺ ∀ ω ∈ Ω \iff \forall \omega\in \Omega ⟺∀ω∈Ω,
ρ ( g h ) ( ω ) = ρ ( g ) ( ρ ( h ) ( ω ) ) o r g h ( ω ) = g ( h ( ω ) ) . \rho(gh)(\omega)=\rho(g)\big(\rho(h)(\omega)\big)\ \ or\ \ gh(\omega)=g\big(h(\omega)\big). ρ(gh)(ω)=ρ(g)(ρ(h)(ω)) or gh(ω)=g(h(ω)).
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轨道: ∀ ω ∈ Ω \forall \omega\in\Omega ∀ω∈Ω, Orb ( ω ) = G ω = [ ω ] = { g ( ω ) ∣ g ∈ G } \text{Orb}(\omega)=G\omega=[\omega]=\{g(\omega)|g\in G\} Orb(ω)=Gω=[ω]={g(ω)∣g∈G}, (类比一个等价类)
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稳定子群: G ω = Sta ( ω ) = { g ∈ G ∣ g ( ω ) = ω } G_\omega=\text{Sta}(\omega)=\{g\in G|g(\omega)=\omega\} Gω=Sta(ω)={g∈G∣g(ω)=ω},
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轨道中元素个数和稳定子群阶数的关系:
G : G ω \] = ∣ G ∣ ∣ G ω ∣ = ∣ G ω ∣ . \[G:G_\\omega\]=\\frac{\|G\|}{\|G_\\omega\|}=\|G\\omega\|. \[G:Gω\]=∣Gω∣∣G∣=∣Gω∣.
例子:
G = S n , Ω = { 1 , 2 , . . . , n } G=\mathcal{S}n,\Omega=\{1,2,...,n\} G=Sn,Ω={1,2,...,n}, 则
G 1 = { g ∣ g ( 1 ) = 1 } ≅ S n − 1 . G_1=\{g|g(1)=1\}\cong \mathcal{S}{n-1}. G1={g∣g(1)=1}≅Sn−1.
例子:
G = Gal ( E / F ) G=\text{Gal}(E/F) G=Gal(E/F),
∀ g ∈ G , g : E → E ( 双射 ) \forall g\in G,g:E\to E\quad (\text{双射}) ∀g∈G,g:E→E(双射)
E E E是 F F F上关于 f f f的分裂域, f ( x ) = ( x − α 1 ) ⋯ ( x − α n ) , Ω = { α 1 , . . . , α n } f(x)=(x-\alpha_1)\cdots(x-\alpha_n), \ \Omega=\{\alpha_1,...,\alpha_n\} f(x)=(x−α1)⋯(x−αn), Ω={α1,...,αn}, G ≲ S E G\lesssim \mathcal{S}{E} G≲SE, 将群作用限制在 Ω \Omega Ω上, 则
G ≲ S Ω = S n , G\lesssim \mathcal{S}{\Omega}=\mathcal{S}_n, G≲SΩ=Sn,
上述的群作用是否仍然是忠实的?(单射) 是.
如果 G G G是 S Ω S_\Omega SΩ上的子群,