Based on paper:
We investigate the Jordan–Hölder property (JHP) in exact categories. First, we introduce a new invariant of exact categories, the Grothendieck monoids, show that (JHP) holds if and only if the Grothendieck monoid is free, and give some numerical criterion. Next, we apply these results to the representation theory of algebras. In most situation, (JHP) holds precisely when the number of projectives is equal to that of simples. We study examples in type A quiver in detail by using combinatorics on symmetric groups.