Jordan-Hölder series
A Jordan-Hölder series of a group 
is a composition series 
of such that 
is a simple group for all integers ![$i\in [0,n-1]$](http://latex.artofproblemsolving.com/1/3/a/13a1ac6785686dc1a366c3c564cbec795199931f.png)
. Equivalently, it is a strictly decreasing composition series of for which there exists no finer strictly decreasing composition series of .
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, 
(the integers mod 4) and the Klein 4-group have equivalent Jordan-Hölder series, but they are not isomorphic.
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