[Paper Review] A note on the number of cyclic subgroups of a finite group

Marius Tărnăuceanu, Mihai-Silviu Lazorec|arXiv (Cornell University)|May 1, 2018

Finite Group Theory Research9 references3 citations

TL;DR

This paper investigates the class 𝒞 of finite nilpotent groups with α(G) = 3/4, where α(G) is the ratio of the number of cyclic subgroups to the group order. It proves that such groups are 2-groups, and fully classifies the non-abelian 2-groups in 𝒞, identifying D₁₆ as the only non-abelian group with a cyclic maximal subgroup in this class, up to elementary abelian 2-group direct factors.

ABSTRACT

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $α(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having $α(G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $\cal{C}$.

Motivation & Objective

• To characterize all finite nilpotent groups satisfying α(G) = 3/4, where α(G) counts cyclic subgroups relative to group order.
• To determine which classes of finite 2-groups—such as generalized dihedral, extraspecial, or those with cyclic maximal subgroups—belong to this class.
• To extend the classification of groups with α(G) > 3/4 to the critical threshold α(G) = 3/4.
• To address the open problem of fully classifying all finite groups with α(G) = 3/4, particularly focusing on 2-groups.

Proposed method

• Use the multiplicative property of α(G) for direct products of groups of coprime order to reduce the problem to p-groups.
• Apply inequalities on the number of cyclic subgroups in p-groups, particularly proving α(G) ≤ α(ℤₚⁿ) for p-groups.
• Leverage known results on subgroup lattices and cyclic subgroup counts in specific 2-groups (e.g., dihedral, quaternion, modular, generalized dihedral).
• Use the formula α(G) = |L₁(G)| / |G| and solve α(G) = 3/4 for specific families of 2-groups by substituting known counts of cyclic subgroups.
• Apply structural theorems on 2-groups with cyclic maximal subgroups (e.g., from [9]) to classify non-abelian cases.
• Use the fact that α(G × ℤ₂ⁿ) = α(G) to reduce classification to groups without elementary abelian 2-group direct factors.

Experimental results

Research questions

• RQ1 Which finite nilpotent groups satisfy α(G) = 3/4?
• RQ2 Which generalized dihedral 2-groups belong to the class 𝒞?
• RQ3 Which non-abelian 2-groups with a cyclic maximal subgroup satisfy α(G) = 3/4?
• RQ4 What is the complete set of finite groups with α(G) = 3/4, and how do they relate to 2-groups and nilpotent structure?
• RQ5 What is the density of the set {α(G) | G finite group} in [0, 3/4]?

Key findings

• The only finite abelian groups in 𝒞 are ℤ₂ⁿ × ℤ₄ for n ∈ ℕ.
• All groups in 𝒞 are 2-groups, and no odd-order p-groups (p odd) satisfy α(G) = 3/4.
• The dihedral group D₁₆ is the only non-abelian 2-group with a cyclic maximal subgroup in 𝒞.
• The generalized dihedral group D(G) is in 𝒞 if and only if G ≅ ℤ₂ⁿ × ℤ₈ for some n ∈ ℕ.
• Among the families of 2-groups with cyclic maximal subgroups, only D₁₆ satisfies α(G) = 3/4.
• Up to direct factors of the form ℤ₂ⁿ, D₁₆ is the unique finite non-abelian 2-group with a cyclic maximal subgroup in 𝒞.