We introduce and investigate subcentral (resp., central) idempotent series and composition subcentral (resp., central) idempotent series in an inverse semigroup. It is shown that if \(S=EG\) is a factorizable inverse monoid with semilattice E of idempotents and the group G of units such that the natural connection \(\theta \) is a dual isomorphism from E to a sublattice of L(G), then any two composition subcentral (resp., central) idempotent series in S are isomorphic. It may be considered as an appropriate analogue in semigroup theory of the Jordan–Hölder Theorem in group theory. Based on this, we also introduce and study G-nilpotent and G-solvable inverse monoids. Some characterizations of the coset monoid of nilpotent groups and solvable groups are given. This extends the main result in Semigroup Forum 20: 255–267 (1980) and also provides another effective approach for the study of nilpotent groups and solvable groups. Finally, some open problems related to nilpotent groups and solvable groups are translated to semigroup theory, which may be helpful for us to solve these open problems.
