[Paper Review] Classification of finite group automorphisms with a large cycle

Alexander Bors|arXiv (Cornell University)|Oct 8, 2014

Finite Group Theory Research17 references4 citations

TL;DR

This paper classifies finite groups $G$ and their automorphisms $\alpha$ for which $\lambda(\alpha) \geq \frac{1}{2}$, where $\lambda(\alpha)$ is the fraction of elements in $G$ lying on a single cycle of the automorphism $\alpha$. Using tools from group theory and permutation cycle structure, the author provides a complete classification of such pairs $(G, \alpha)$ up to isomorphism, identifying all finite groups admitting an automorphism with a cycle covering at least half the group elements.

ABSTRACT

Let $ψ$ be a permutation of a finite set $X$. We define $λ(ψ)$ to be the largest fraction of elements of $X$ lying on a single cycle of $ψ$. For a finite group $G$, we define $λ(G)$ to be the maximum among the values $λ(α)$, where $α$ runs through the automorphisms of $G$. In this paper, we develop tools to deal with questions related to $λ$-values of finite groups and of their automorphisms. As a consequence, we will be able to give a classification, up to a natural notion of isomorphism, of those pairs $(G,α)$ where $G$ is a finite group, $α$ is an automorphism of $G$ and $λ(α)\geq\frac{1}{2}$.

Motivation & Objective

• To classify all finite groups $G$ and automorphisms $\alpha$ of $G$ for which $\lambda(\alpha) \geq \frac{1}{2}$, where $\lambda(\alpha)$ measures the largest cycle fraction under $\alpha$.
• To extend existing results on automorphism cycle structures by focusing on the maximal cycle length relative to group size.
• To provide a complete classification of such group-automorphism pairs up to isomorphism, rather than just classifying groups.
• To explore connections between automorphism cycle structure and pseudorandom number generation, particularly in the context of finite dynamical systems.
• To lay groundwork for future study of automorphism and affine large-cycle conditions for $\rho < \frac{1}{2}$.

Proposed method

• Define $\lambda(\alpha)$ as the fraction of elements in a finite group $G$ lying on the longest cycle of an automorphism $\alpha \in \mathrm{Aut}(G)$.
• Use structural group theory, including properties of nilpotent and abelian groups, to analyze the maximum possible cycle length of automorphisms.
• Leverage known results on automorphism orders in $p$-groups and elementary abelian groups, particularly the fact that $\lambda((\mathbb{Z}/p\mathbb{Z})^k) = 1 - \frac{1}{p^k}$.
• Apply Horoševskiï’s theorem that $|G| - 1$ is an upper bound on automorphism cycle length for nontrivial $G$.
• Use combinatorial and number-theoretic estimates (e.g., product bounds on cycle lengths) to rule out certain configurations via contradiction.
• Classify all isomorphism types of pairs $(G, \alpha)$ with $\lambda(\alpha) \geq \frac{1}{2}$, including non-abelian examples such as dihedral and semidihedral groups with specific automorphisms.

Experimental results

Research questions

• RQ1 Which finite groups $G$ admit an automorphism $\alpha$ such that at least half of the elements lie on a single cycle of $\alpha$?
• RQ2 What is the complete isomorphism-class list of pairs $(G, \alpha)$ where $\lambda(\alpha) \geq \frac{1}{2}$?
• RQ3 How do the cycle structures of automorphisms in non-abelian groups compare to those in abelian or elementary abelian groups in terms of cycle length fraction?
• RQ4 Can the classification of large-cycle automorphisms be extended to cases where $\rho < \frac{1}{2}$?
• RQ5 What is the role of automorphism cycle structure in finite dynamical systems used for pseudorandom number generation?

Key findings

• The paper provides a complete classification of all isomorphism types of pairs $(G, \alpha)$ where $G$ is a finite group and $\alpha \in \mathrm{Aut}(G)$ satisfies $\lambda(\alpha) \geq \frac{1}{2}$.
• Elementary abelian $p$-groups $(\mathbb{Z}/p\mathbb{Z})^k$ achieve $\lambda$-values of $1 - \frac{1}{p^k}$, and for $k \geq 1$, these can exceed $\frac{1}{2}$ when $p^k \geq 3$.
• For $p$-groups, the $\lambda$-value of $\mathbb{Z}/p^k\mathbb{Z}$ is $1 - \frac{1}{p}$, so for odd $p$, this exceeds $\frac{1}{2}$, while for $p=2$, $\lambda(\mathbb{Z}/2^k\mathbb{Z}) = \frac{1}{4}$ for $k \geq 3$.
• Non-abelian groups such as dihedral groups $D_{2n}$ and certain semidihedral groups admit automorphisms with $\lambda(\alpha) \geq \frac{1}{2}$, including cases where $\lambda(\alpha) > \frac{18}{19} \cdot \prod_{n=3,4,5} (1 - \frac{1}{2^n}) \cdot \mathrm{exp}(-\frac{1}{2^9}) > 0.751$.
• The classification shows that apart from classical cyclic and elementary abelian groups, no fundamentally new finite groups admit automorphisms with large cycles when $\lambda(\alpha) \geq \frac{1}{2}$.
• The results imply that for pseudorandom number generation, if a large cycle fraction $\geq \frac{1}{2}$ is required, only known classical groups (cyclic and elementary abelian) or their automorphism-induced systems are viable, with no new types emerging.