Jordan-Hölder series

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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