Finite groups with nearly half as many cyclic subgroups as elements

2025/06/26 by Chhajer, Vaibhav, Hatui, Sumana, Sharma, Palash · 1 citation
#20D25 #20E34 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2506.21163

Abstract

Suppose C(G) denotes the set of all cyclic subgroups of a finite group G, and O₂(G) denotes the number of elements of order 2 in G. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American Mathematical Monthly 122.3 (2015): 275-276], an open problem was asked to classify the groups G with |C(G)|=|G|-r, where 2 ≤ r ≤ |G|-1. In this article, first we show that, for an odd prime p, there are infinitely many groups G with |C(G)|= (|G|)/(2), |C(G)|=\frac|G|p^q-1 (for prime q≠ p), or |C(G)|=(|G|)/(2)+2^k, k≥ 0. Then, we partially answer the open question by classifying finite groups G having (|G|)/(2)-1≤ |C(G)| ≤ (|G|)/(2)+1 for some fix values of O₂(G). Finally, we provide a complete list of finite groups G having |C(G)|=(|G|+(2r+1))/(2) for r≥-1.

Citations

• Finite Groups with a Prescribed Number of Cyclic Subgroups II
• A note on the number of cyclic subgroups of a finite group
• Finite groups with small number of cyclic subgroups

Cited by

• Finite groups with mostly involuted cyclic subgroups

Related

• Finite groups with a certain number of cyclic subgroups II
• Finite Groups with a Prescribed Number of Cyclic Subgroups II
• Finite groups with mostly involuted cyclic subgroups
• Finite groups with specific number of cyclic subgroups
• Finite Groups with a Prescribed Number of Cyclic Subgroups