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单词 Jordan Holder theorem
释义

Jordan Holder theorem

中文百科

合成列 Composition series

(重定向自Jordan Holder theorem)

在抽象代数中。合成列是借着将代数对象(如群、模等等)拆解为简单的成份,以萃取不变量的方式之一。以模为例,一般环上的模未必能表成单模的直和。但是我们可退而求其次,考虑一组过滤 \{0\}= M_0 \subset \cdots \subset M_n=M,使每个子商 M_i/M_{i+1} 皆为单模;这些单模称为合成因子n 称为合成长度,都是 M 的不变量。亦可考虑 M 的子模范畴 \mathcal{A},此时 [M] \in K(\mathcal{A}) 可唯一表为合成因子之和;在此意义下,K-群提供了模的半单化

合成列未必存在,即使存在也未必唯一。然而若尔当-赫尔德定理断言:若一对象有合成列,则子商的同构类是唯一确定的,至多差一个置换。因此,合成列给出有限群或阿廷模的不变量。

英语百科

Composition series 合成列

(重定向自Jordan Holder theorem)

In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.

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