Jordan-Hölder decomposition theorem

Every finite group $G$ has a filtration

$$G\supset G_0\supset \mathrm{\cdots }\supset G_n=\{1\},$$

where each $G_{i+1}$ is normal in $G_i$ and each quotient group $G_i/G_{i+1}$ is a simple group. Any two such decompositions of $G$ have the same multiset of simple groups $G_i/G_{i+1}$ up to ordering.

A filtration of $G$ satisfying the properties above is called a Jordan–Hölder decomposition of $G$.

Title	Jordan-Hölder decomposition theorem
Canonical name	JordanHolderDecompositionTheorem
Date of creation	2013-03-22 12:08:44
Last modified on	2013-03-22 12:08:44
Owner	djao (24)
Last modified by	djao (24)
Numerical id	9
Author	djao (24)
Entry type	Theorem
Classification	msc 20E32
Related topic	SubnormalSeries
Defines	Jordan-Hölder decomposition