Cyclicity degrees of finite groups

TL;DR

This paper introduces the cyclicity degree of finite groups, quantifies the probability that a random subgroup is cyclic, and provides formulas, asymptotic behavior, and extremal results for various group classes.

Contribution

It defines and analyzes the cyclicity degree, offering explicit formulas, asymptotic analysis, and extremal properties for specific finite groups.

Findings

Explicit formulas for cyclicity degrees in certain groups

Asymptotic behavior of cyclicity degrees as group size grows

Minimality and maximality results for cyclicity degrees

Abstract

We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An asymptotic formula and several minimality/maximality results on cyclicity degrees are also inferred.