# 合成列

（重定向自約當-赫德定理

## 群的情形

${\displaystyle G}$  為群，${\displaystyle G}$  的合成列是對應於一族子群

${\displaystyle \{e\}=H_{0}\subset H_{1}\subset \cdots \subset H_{n}=G}$

## 模的情形

${\displaystyle \{0\}=J_{0}\subset \cdots \subset J_{n}=M}$

## 例子

${\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{6}\triangleleft C_{12}}$ ,
${\displaystyle C_{1}\triangleleft C_{2}\triangleleft C_{4}\triangleleft C_{12}}$ ,
${\displaystyle C_{1}\triangleleft C_{3}\triangleleft C_{6}\triangleleft C_{12}}$

${\displaystyle C_{2},C_{3},C_{2}}$
${\displaystyle C_{2},C_{2},C_{3}}$
${\displaystyle C_{3},C_{2},C_{2}}$

## 若尔当-赫尔德定理

${\displaystyle \{0\}=M_{0}\subset \cdots \subset M_{r}=M}$
${\displaystyle \{0\}=M'_{0}\subset \cdots \subset M'_{s}=M}$

${\displaystyle \mathrm {min} (r,s)}$ 數學歸納法。若 ${\displaystyle \mathrm {min} (r,s)=0}$ ${\displaystyle M=0}$ ，若 ${\displaystyle \mathrm {min} (r,s)=1}$ ${\displaystyle M}$ 單模。以下假定 ${\displaystyle r,s\geq 2}$

${\displaystyle M_{r-1}=M_{s-1}}$ ，據歸納法假設，${\displaystyle r-1=s-1}$ ${\displaystyle M_{i+1}/M_{i}}$ ${\displaystyle M'_{i+1}/M'_{i}}$ ${\displaystyle 0\leq i\leq r-2}$ ）之間僅差置換。此外 ${\displaystyle M/M_{r-1}=M_{/}M'_{s-1}}$ ，故定理成立。

${\displaystyle M_{r-1}\neq M'_{s-1}}$ 。此時必有 ${\displaystyle M_{r-1}+M'_{s-1}=M}$ 。置 ${\displaystyle N:=M_{r-1}\cap M'_{s-1}}$ ，於是

${\displaystyle M/M_{r-1}=(M_{r-1}+M'_{s-1})/M_{r-1}\simeq M'_{s-1}/N}$
${\displaystyle M/M'_{s-1}=(M_{r-1}+M'_{s-1})/M'_{s-1}\simeq M_{r-1}/N}$

${\displaystyle N}$  的合成列 ${\displaystyle \{0\}=K_{0}\subset \cdots \subset K_{t}=N}$ ，依上式知

${\displaystyle \{0\}=K_{0}\subset \cdots \subset K_{t}=N\subset M_{r-1}\subset M\quad \ldots (*)}$
${\displaystyle \{0\}=K_{0}\subset \cdots \subset K_{t}=N\subset M'_{s-1}\subset M\quad \ldots (**)}$