Bounds in terms of the 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.