Bounds in terms of the number of cyclic subgroups

TL;DR

This paper introduces new classifications of groups based on the finiteness of groups with a given number of cyclic subgroups, providing characterizations of prime numbers through group properties.

Contribution

It establishes the concepts of cyclic bounded and maximal cyclic bounded families, and characterizes prime numbers via finiteness conditions on noncyclic groups.

Findings

Family of groups of prime power order is maximal cyclic bounded.

Family of finite groups without cyclic coprime direct factors is cyclic bounded.

Prime numbers are characterized by finiteness of noncyclic groups with a specific number of cyclic subgroups.

Abstract

A family of groups is called (maximal) cyclic bounded ((M)CB) if, for every natural number $n$, there are only finitely many groups in the family with at most $n$ (maximal) cyclic subgroups. We prove that the family of groups of prime power order is MCB. We also prove that the family of finite groups without cyclic coprime direct factors is CB. As a consequence, a natural number $n \geqslant 10$ is prime if and only if there are only finitely many finite noncyclic groups with precisely $n$ cyclic subgroups.