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 |G|-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 |G|-∆ cyclic subgroups for Δ = 2, 3, 4 and 5. In this paper, we prove that for any Δ > 0 if G has exactly |G|-∆ cyclic subgroups, then |G|-≤ 8∆ and therefore the number of such G is finite. We then use the computer program GAP to find all G with exactly |G|-∆ cyclic subgroups for ∆ = 1,..., 32.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1080/00927872.2019.1572172

Keywords

Cyclic subgroups; finite group theory; structure of finite groups

Publication Date

10-3-2019

Journal Title

Communications in Algebra

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