Jordan-Hölder series
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.
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