Addendum to "Finite groups with a prescribed number of cyclic subgroups". Issue 10 (3rd October 2019)

• Record Type: Journal Article
• Authors: Belshoff, Richard; Dillstrom, Joe; Reid, Les
• Is Part Of: Communications in algebra. Volume 47:Issue 10(2019)
• Page Start: 3939
• Page End: 3940
• Publication Date: 2019-10-03
• Subjects: Cyclic subgroups; finite group theory; structure of finite groups; 20D99; 20E07; 20E34; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2019.1572172
• Languages: English
• ISSNs: 0092-7872

Abstract

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