# Jordan-Hölder decomposition theorem

Every finite group $G$ has a filtration

 $G\supset G_{0}\supset\cdots\supset G_{n}=\{1\},$

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 JordanHolderDecompositionTheorem 2013-03-22 12:08:44 2013-03-22 12:08:44 djao (24) djao (24) 9 djao (24) Theorem msc 20E32 SubnormalSeries Jordan-Hölder decomposition