Jordan-H\"older decomposition theoremJordanHolderDecompositionTheorem2002-01-05 03:35:562004-06-23 01:48:08TheoremJordan-H\"older decomposition\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{xypic}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 {\em Jordan--H\"older decomposition} of $G$.