I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite groups using the Jordan-Holder theorem in the case of group algebras. Thanks in advance.
2026-03-25 22:09:59.1774476599
Jordan-Holder theorem for group algebras
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I don't think so, due to non-commutativity of groups and the fact that every module has an underlying abelian group. That's not a strict proof of impossibility, but it seems unlikely.
However, there is a way to prove a version of Jordan-Hölder that covers finite groups, finite length modules over a ring (and more!): you need to introduce groups with operators, which simultaneously generalize groups and modules over a ring. IIRC, Jacobson treats this in Basic Algebra I.