On the number of cyclic transitive subgroups of a permutation group

TL;DR

This paper establishes an upper limit on the number of cyclic transitive subgroups in finite permutation groups, explores the structure of groups reaching this limit, and applies findings to number field theory.

Contribution

It provides a new upper bound for cyclic transitive subgroups and characterizes groups that attain this bound, with applications to number fields.

Findings

Derived an upper bound for cyclic transitive subgroups

Characterized groups achieving the bound

Applied results to number field theory

Abstract

We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.