Finite Groups with a Prescribed Number of Cyclic Subgroups II

2018/10/19 by Richard Belshoff, Belshoff, Richard, Joe Dillstrom +3 · 2 citations
Mathematics · #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1810.08328

Abstract

Tărnăuceanu described the finite groups G having exactly |G|-1 cyclic subgroups. In "Finite Groups with a Prescribed Number of Cyclic Subgroups,", we 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|≤ 8Δ 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.

Cited by

• Finite groups with mostly involuted cyclic subgroups
• Finite groups with nearly half as many cyclic subgroups as elements

Related

• Finite Groups with a Prescribed Number of Cyclic Subgroups
• Finite groups with specific number of cyclic subgroups
• Finite groups with a certain number of cyclic subgroups II
• On the number of cyclic subgroup in finite groups
• Finite groups with nearly half as many cyclic subgroups as elements