[Paper Review] On some probabilistic aspects of (generalized) dicyclic groups

Marius Tărnăuceanu, Mihai-Silviu Lazorec|arXiv (Cornell University)|Dec 6, 2016

Finite Group Theory Research4 references3 citations

TL;DR

This paper derives explicit formulas for the subgroup commutativity degree (sd) and cyclic subgroup commutativity degree (csd) of dicyclic and generalized dicyclic groups, particularly $\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_n)$, and proves that $\mathrm{csd}(\mathbb{Z}_2 \times Q_{2^{m+1}}) \to 0$ as $m \to \infty$, establishing a new class of groups with asymptotically vanishing cyclic subgroup commutativity degree.

ABSTRACT

In this paper we study probabilistic aspects such as subgroup commutativity degree and cyclic subgroup commutativity degree of the (generalized) dicyclic groups. We find explicit formulas for these concepts and we provide another example of a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.

Motivation & Objective

• Investigate the probabilistic structure of dicyclic and generalized dicyclic groups through subgroup commutativity degree (sd) and cyclic subgroup commutativity degree (csd).
• Provide explicit closed-form expressions for sd and csd in $\mathrm{Dic}_{4n}$ and $\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_n)$, focusing on $n = 2^m$.
• Analyze the asymptotic behavior of csd, particularly for $\mathbb{Z}_2 \times Q_{2^{m+1}}$, to determine whether it tends to zero.
• Extend existing results on (cyclic) subgroup commutativity degrees to a new class of non-abelian groups with non-trivial structure.
• Identify open problems for future research on the interplay between group structure and probabilistic invariants like sd and csd.

Proposed method

• Use the group presentation $\mathrm{Dic}_{4n} = \langle a, \gamma \mid a^{2n} = e, \gamma^2 = a^n, a^\gamma = a^{-1} \rangle$ to classify subgroups and cyclic subgroups.
• Leverage subgroup lattice structure: $\tau(2n)$ cyclic subgroups of $\langle a \rangle$, and $\sigma(n)$ dicyclic subgroups $H_i^s$ isomorphic to $\mathrm{Dic}_{4s}$.
• Apply the definition $sd(G) = \frac{1}{|L(G)|^2} \sum_{H \in L(G)} |C(H)|$, where $C(H)$ is the set of subgroups commuting with $H$.
• For csd, compute $csd(G) = \frac{1}{|L_1(G)|^2} \sum_{H \in L_1(G)} |C_1(H)|$, focusing on cyclic subgroups and their commutativity relations.
• Use group isomorphisms such as $\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_n) \cong \mathbb{Z}_2 \times Q_{2^{m+1}}$ when $n = 2^m$ to simplify computation.
• Employ number-theoretic functions: $\tau(m')$ (number of divisors), $\sigma(n)$ (sum of divisors), and a custom arithmetic function $g(m')$ for odd $m'$.

Experimental results

Research questions

• RQ1 What is the explicit formula for the subgroup commutativity degree $sd(\mathrm{Dic}_{4n})$ in terms of the prime factorization of $n$?
• RQ2 How does the cyclic subgroup commutativity degree $csd(\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_n))$ behave asymptotically as $n = 2^m$ grows?
• RQ3 What is the structure of the subgroup lattice and cyclic subgroup poset in generalized dicyclic groups $\mathrm{Dic}_{4n}(A)$ for $A \cong \mathbb{Z}_2 \times \mathbb{Z}_n$?
• RQ4 Does $csd(\mathbb{Z}_2 \times Q_{2^{m+1}})$ tend to zero as $m \to \infty$, and if so, what does this imply about the probabilistic non-commutativity of cyclic subgroups?
• RQ5 Can explicit formulas for $sd$ and $csd$ be derived for $\mathrm{Dic}_{4n}(A)$ when $A$ is an arbitrary abelian group of order $2n$?

Key findings

• The subgroup commutativity degree of $\mathrm{Dic}_{4n}$ is given by $sd(Dic_{4n}) = \frac{(m+2)^2\tau(m')^2 + 2(m+2)\tau(m')\sigma(n) + [(m-1)2^{m+3}+9]g(m')}{[(m+2)\tau(m') + \sigma(n)]^2}$, where $n = 2^m m'$, $m' \in \mathbb{N}$ odd, and $g(m')$ is a multiplicative arithmetic function.
• For $\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_n)$ with $n = 2^m$, the cyclic subgroup commutativity degree is $csd(\mathbb{Z}_2 \times Q_{2^{m+1}}) = \frac{2^m(4m+8) + (2m+2)^2}{(2^m + 2m + 2)^2}$.
• The cyclic subgroup commutativity degree of $\mathbb{Z}_2 \times Q_{2^{m+1}}$ tends to zero as $m \to \infty$, i.e., $\lim_{m \to \infty} csd(\mathbb{Z}_2 \times Q_{2^{m+1}}) = 0$.
• Explicit values include $csd(\mathbb{Z}_2 \times Q_8) = 1$, $csd(\mathbb{Z}_2 \times Q_{16}) = \frac{224}{256} = \frac{7}{8}$, and $csd(\mathbb{Z}_2 \times Q_{32}) \to 0$.
• The cyclic subgroup commutativity degree of $\mathrm{Dic}_{4n}(\mathbb{Z}_2 \times \mathbb{Z}_6)$ is $\frac{43}{49}$, showing non-trivial intermediate behavior.
• The paper provides a new class of finite non-abelian groups—specifically $\mathbb{Z}_2 \times Q_{2^{m+1}}$—for which the cyclic subgroup commutativity degree vanishes asymptotically.