Probabilistic aspects of ZM-groups

TL;DR

This paper investigates probabilistic properties of ZM-groups, including subgroup and factorization degrees, demonstrating their computation via conjugacy class sizes and identifying groups with asymptotically vanishing cyclic subgroup commutativity degree.

Contribution

It introduces formulas for probabilistic measures in ZM-groups based on conjugacy class sizes and explores asymptotic behavior of these measures in certain groups.

Findings

Probabilistic measures can be computed using conjugacy class sizes.

Identifies groups with cyclic subgroup commutativity degree that vanishes asymptotically.

Provides explicit formulas for subgroup and factorization degrees in ZM-groups.

Abstract

In this paper we study probabilistic aspects such as (cyclic) subgroup commutativity degree and (cyclic) factorization number of ZM-groups. We show that these quantities can be computed using the sizes of the conjugacy classes of these groups. In the end, we point out a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.