Addendum to “Finite groups with a prescribed number of cyclic subgroups”
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.
Cyclic subgroups; finite group theory; structure of finite groups
Belshoff, Richard, Joe Dillstrom, and Les Reid. "Addendum to 'Finite groups with a prescribed number of cyclic subgroups'." Communications in Algebra 47, no. 10 (2019): 3939-3940.
Communications in Algebra