Addendum to “Finite groups with a prescribed number of cyclic subgroups”

Abstract

In [Tărnăuceanu, M. (2015). Finite groups with a certain number of cyclic subgroups. Amer. Math. Monthly. 122:275–276], Tărnăuceanu described the finite groups G having exactly $\left|G\right|-1$ cyclic subgroups. In [Belshoff, R., Dillstrom, J., Reid, L. Finite groups with a prescribed number of cyclic subgroups. To appear in Communications in Algebra], the authors used elementary methods to completely characterize those finite groups G having exactly $\left|G\right|-\Delta$ cyclic subgroups for Δ = 2, 3, 4 and 5. In this paper, we prove that for any Δ > 0 if G has exactly $\left|G\right|-\Delta$ cyclic subgroups, then $\left|G\right|\le 8\Delta$ and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly $\left|G\right|-\Delta$ cyclic subgroups for $\Delta =1,\dots ,32$.

Keywords:

• Cyclic subgroups; finite group theory; structure of finite groups

MATHEMATICS SUBJECT CLASSIFICATION:

• 20D99; 20E07; 20E34

Acknowledgement

This paper is an addendum to [1], which was an expansion and revision of the Master’s thesis of the second author, directed by the first and third authors.