If I have two finite groups $H$ and $G$ where $|G| = |H|$, my guess will be that their composition series don't necessarily have the same length. But I'm not able to find a counterexample. Where should I search?
2026-04-24 02:11:26.1776996686
Do two finite groups of the same order have composition series of the same length?
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What about $S_5$ and $\mathbb{Z}/{120\mathbb{Z}}$?
Or even easier, $A_5$ and $\mathbb{Z}/60\mathbb{Z}$. Take any non-abelian simple group $G$ so its composition series has length one, and then compare that to the length of the composition series of the cyclic group of size $|G|$.