Groups having 12 cyclic subgroups

TL;DR

This paper classifies all finite groups with exactly 12 cyclic subgroups and shows that cyclicity degrees of finite groups are dense in the interval [0,1], addressing a question about possible limit values.

Contribution

It provides a complete classification of 12-cyclic groups and proves the density of cyclicity degrees in [0,1], solving an open problem in the field.

Findings

Classified all 12-cyclic groups.

Proved cyclicity degrees are dense in [0,1].

Addressed an open problem by Trnuceanu and Tth.

Abstract

A finite group is said to be $n$-cyclic if it contains $n$ cyclic subgroups. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group $G$ and is denoted by $cdeg (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which gives a solution to the problem asked by T\u{a}rn\u{a}uceanu and T\'{o}th in [20] "For every $a\in [0, 1]$, does there exist a sequence $(G_n)$ of finite groups such that $\lim_{n\to\infty} cdeg(G_n)=a$ "?