##  [Jordan–Hölder Theorem](/index.php/jordan-holder-theorem) 

  ##  [Jordan–Hölder Theorem](https://mathlogic.quantumdictionary.io/jordan-holder-theorem-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/index.php/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A theorem stating that for any finite-length group or module, any two composition series have isomorphic simple factor modules up to reordering; equivalently, the multiset of composition factors (simple quotients) and the composition length are invariants of the object.

 

 

 

 

 





 

 



 ##  [Jordan–Hölder Theorem](https://puremath.quantumdictionary.io/jordan-holder-theorem-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/index.php/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The theorem that any two composition series of a finite-length object (for example a finite group or a finite-length module) have isomorphic multisets of simple composition factors, possibly in different orders; the multiset of factors is therefore well-defined.