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群同态

数学 群论入门 · 08

上一章末尾提到的陪集置换表示SGHS_{GH}就是GG的同态

同态的定义

同构满足双射

同态是更一般的映射

设φ\varphi是GG到HH的同态映射

∀a,b∈G,(ab)φ=aφbφ  ⟺  G∼H\forall a, b \in G, (ab)^{\varphi} = a^{\varphi}b^{\varphi} \iff G \sim H

同态性质

  1. eφ=e′e^{\varphi} = e'

    证明：

    (eφ)2=eφ=eφe′ (e^{\varphi})^2 = e^{\varphi} = e^{\varphi}e'

    eφ=e′e^{\varphi} = e'

  2. (a−1)φ=(aφ)−1(a^{-1})^{\varphi} = (a^{\varphi})^{-1}

  3. ord(a)=k  ⟹  ord(aφ)∣kord(a) = k \implies ord(a^{\varphi}) \mid k

    证明：

    设aφa^{\varphi}的阶mm

    (aφ)k=(ak)φ=eφ=e′(a^{\varphi})^k = (a^k)^{\varphi} = e^{\varphi} = e'

    得m∣km \mid k

  4. ∀a,b∈G,ab=ba  ⟹  aφbφ=bφaφ\forall a,b \in G, ab = ba \implies a^{\varphi}b^{\varphi} = b^{\varphi}a^{\varphi}

  5. ∀A≤G,Aφ≤H\forall A \le G, A^{\varphi} \le H

  6. B≤HB \le H，记BB的逆象Bφ−1B^{\varphi^{-1}}：

    {a∣aφ∈B}\{a \mid a^{\varphi} \in B\}

    有Bφ−1≤GB^{\varphi^{-1}} \le G

同态的核

定义B={e′}B = \{e'\}的逆象为φ\varphi的核。记作ker⁡φ\ker \varphi

定理

  1. φ是满同态映射，且ker⁡φ={e}  ⟺  φ是同构映射\varphi是满同态映射，且\ker \varphi = \{e\} \iff \varphi是同构映射

  2. aφ=bφ  ⟺  a,b∈Kxa^{\varphi} = b^{\varphi} \iff a,b \in Kx