Jordan-Holder theorem for imprimitivity systems and maximal decompositions of rational functions

TL;DR

This paper explores the structure of imprimitivity systems in permutation groups with cyclic subgroups, generalizes Ritt's theorem on polynomial decompositions, and examines cases where classical results do not hold for rational functions.

Contribution

It proves new results about imprimitivity systems containing cyclic subgroups and extends Ritt's theorem to broader classes of rational functions.

Findings

Generalized Ritt's theorem for polynomial decompositions

Identified conditions where Ritt's theorem fails for rational functions

Analyzed the lattice of imprimitivity systems in specific permutation groups

Abstract

In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional decompositions of polynomials, and some other related results. Besides, we discuss examples of rational functions, related to finite subgroups of the automorphism group of the sphere for which the first Ritt theorem fails to be true.