Is there some generalization of the Jordan-Hölder decomposition for group objects in a category $\mathcal{C}$?

If $\mathcal{C}$ is the category Sch$(S)$ of schemes over a base scheme $S$ then (I think) this is true, also probably for other categories of "spaces" like Top or Diff it should be true, but I don't have any idea for general categories.

  • $\begingroup$ In a general category, it's not a priori true that even the isomorphism theorems will hold. For an example, see this answer by JS Milne about classical reductive groups. If these "theorems" don't hold, I think it's pretty clear that you can't prove Jordan-Hölder. $\endgroup$ Mar 20, 2010 at 13:01
  • $\begingroup$ Could you be more specific as to what you mean by the Jordan-Holder decomposition? Do you mean the uniqueness of composition factors (as multisets) in any two composition series? (Of course any kind of statement about existence of composition series is not going to be true even for general group objects in the category of sets.) $\endgroup$ Mar 20, 2010 at 16:12
  • $\begingroup$ That's exactly what I meant, given that there exists a composition series, it should be unique up to permutation. After thinkingof it for a while, I believe that it should be true for any category in which the second isomorphism theorem holds. $\endgroup$ Mar 20, 2010 at 17:02
  • $\begingroup$ Which one is the second iso thm? It varies according to who you ask. $\endgroup$ Mar 21, 2010 at 2:41
  • $\begingroup$ I mean the following: if $S$ is a subgroup and $N$ a normal subgroup of $G$ then $N$ is normal in $NS$, $N\cap S$ is normal in $S$ and $NS/N$ is isomorphic to $S/N\cap S$. $\endgroup$ Mar 22, 2010 at 18:34

1 Answer 1


You can start by looking at the paper by P.J. Hilton and W. Ledermann, "On the Jordan-Hölder theorem in homological monoids", where three axioms are needed to establish the decomposition, and the third one is essentially guaranteeing the second isomorphism theorem.

In "Mal'cev, protomodular, homological and semi-abelian categories" by F. Borceux, D. Bourn, there is a chapter devoted to homological categories (which are pointed, regular and protomodular), these are the categories where certain lemmas of homological algebra hold true (five lemma, nine lemma, snake lemma, Noether isomorphism theorems etc.). The fact that Jordan-Holder holds for these categories is proven in "Jordan-Holder, Modularity and Distributivity in Non-Commutative Algebra" by F. Borceux, M. Grandis.


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