Jordan-Hölder series

Revision as of 12:19, 10 May 2008 by Boy Soprano II (talk | contribs) (New page: A '''Jordan-Hölder series''' of a group <math>G</math> is a composition series <math>(G_i)_{0\le i \le n}</math> of <math>G</math> such that <math>G_i/G_{i+1}</math> is a [[simple...)
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A Jordan-Hölder series of a group $G$ is a composition series $(G_i)_{0\le i \le n}$ of $G$ such that $G_i/G_{i+1}$ is a simple group for all integers $i\in [0,n-1]$. Equivalently, it is a strictly decreasing composition series of $G$ for which there exists no finer strictly decreasing composition series of $G$.

The Jordan-Hölder Theorem says that any two Jordan-Hölder series of the same group are equivalent. Unfortunately, non-isomorphic groups can have equivalent Jordan-Hölder series. For instance, $\mathbb{Z}/4\mathbb{Z}$ (the integers mod 4) and the Klein 4-group have equivalent Jordan-Hölder series, but they are not isomorphic.

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