# 內自同構

## 定義

{\displaystyle {\begin{aligned}\varphi _{g}\colon G&\longrightarrow G\\\varphi _{g}(x)&:=g^{-1}xg\end{aligned}}}

${\displaystyle \varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=(g^{-1}x_{1}g)(g^{-1}x_{2}g)=\varphi _{g}(x_{1})\varphi _{g}(x_{2}),}$

## 內外自同構羣

${\displaystyle \operatorname {Out} (G)=\operatorname {Aut} (G)/\operatorname {Inn} (G).}$

${\displaystyle a^{-1}xa=x\iff ax=xa.}$

${\displaystyle G/Z(G)\cong \operatorname {Inn} (G).}$

### 有限p-羣嘅非內自同構

Wolfgang Gaschütz嘅一個定理話如果${\displaystyle G}$係一個有限唔交換p-羣嘅話，噉${\displaystyle G}$就有一個非內自同構嘅階係p嘅次方。

1. ${\displaystyle G}$係二階零冪
2. ${\displaystyle G}$regular p-羣
3. ${\displaystyle G/Z(G)}$powerful p-羣
4. ${\displaystyle G}$Frattini子羣嘅中心嘅中心化子CGZ ∘ Φ(G)，同Φ(G)唔一樣

### 羣嘅種類

${\displaystyle G}$嘅內自同構羣${\displaystyle \operatorname {Inn} (G)}$係平凡若且唯若${\displaystyle G}$交換羣，而${\displaystyle \operatorname {Inn} (G)}$如果係循環羣嘅話就只能係平凡羣。

## 參考

1. S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (第3.版). Hoboken, NJ: Wiley. p. 45. ISBN 9780471452348. OCLC 248917264.
2. Schupp, Paul E. (1987), "A characterization of inner automorphisms" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 101 (2): 226–228, doi:10.2307/2045986, JSTOR 2045986, MR 0902532