A Jordan-Hölder series of a group is a composition series of such that is a simple group for all integers . 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.
This article is a stub. Help us out by.