[Paper Review] Cycle lengths in finite groups and the size of the solvable radical

Alexander Bors|arXiv (Cornell University)|Jan 28, 2015

Finite Group Theory Research6 references3 citations

TL;DR

This paper establishes sharp bounds on the size of the solvable radical in finite groups based on the existence of long cycles in automorphisms or periodic affine maps. It proves that if a finite group has an automorphism with a cycle of length at least $\rho|G|$, then the index $[G: \operatorname{Rad}(G)]$ is bounded in terms of $\rho$, and solvability is implied if $\rho > \frac{1}{10}$; similarly, for cycle lengths of order $|G|^e$, solvability is implied when $e > \frac{1}{3}$ for automorphisms and $e > \frac{2}{3}$ for affine maps.

ABSTRACT

We prove the following: For any $ρ\in\left(0,1 ight)$, if a finite group $G$ has an automorphism with a cycle of length at least $ρ\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $ρ$, and such a condition is strong enough to imply solvability of $G$ if and only if $ρ>\frac{1}{10}$. Furthermore, considering, for exponents $e\in\left(0,1 ight)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\operatorname{Rad}(G)| o\infty$ for $|G| o\infty$ if and only if $e>\frac{1}{3}$. We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.

Motivation & Objective

• To determine how the existence of long cycles in automorphisms or periodic affine maps constrains the structure of finite groups.
• To quantify the relationship between cycle length and the size of the solvable radical $\operatorname{Rad}(G)$ in finite groups.
• To identify sharp thresholds $\rho$ and $e$ such that the presence of long cycles implies solvability of $G$.
• To extend results from automorphisms to a broader class of maps—periodic affine maps—while maintaining structural constraints.
• To provide explicit bounds on $[G: \operatorname{Rad}(G)]$ under cycle length conditions, using the classification of finite simple groups (CFSG).

Proposed method

• Leverages the classification of finite simple groups (CFSG) to analyze maximum cycle lengths of automorphisms and periodic affine maps on finite nonabelian characteristically simple groups.
• Uses technical lemmas to bound the maximum order of elements in automorphism groups of finite simple groups of Lie type.
• Applies asymptotic estimates and inequalities involving group orders and cycle lengths, particularly for classical and exceptional groups of Lie type.
• Employs case analysis by group type (e.g., $\operatorname{PSL}_d(q)$, $\operatorname{PSU}_d(q)$, symplectic, orthogonal, exceptional groups), using known bounds on element orders and outer automorphism group sizes.
• Introduces auxiliary functions $g(n)$ and constants $K(\epsilon,\xi)$, $K_{\mathrm{aff}}(\epsilon,\xi)$ to control error terms in asymptotic bounds.
• Reduces the main results to a single technical lemma (Lemma 3.1) concerning maximum cycle lengths in finite nonabelian characteristically simple groups.

Experimental results

Research questions

• RQ1 For which values of $\rho \in (0,1)$ does the existence of an automorphism with a cycle of length at least $\rho|G|$ imply that $G$ is solvable?
• RQ2 How does the presence of a periodic affine map with a long cycle affect the index $[G: \operatorname{Rad}(G)]$?
• RQ3 What is the threshold exponent $e$ such that a cycle length of at least $|G|^e$ in an automorphism forces $|\operatorname{Rad}(G)| \to \infty$ as $|G| \to \infty$?
• RQ4 Can the bound on $[G: \operatorname{Rad}(G)]$ be uniformly controlled under cycle length conditions of the form $|G|^e$?
• RQ5 Are there infinite families of non-solvable groups with automorphisms or periodic affine maps having cycle lengths exceeding $|G|^{1/3}$ or $|G|^{2/3}$ respectively?

Key findings

• If a finite group $G$ has an automorphism with a cycle of length greater than $\frac{1}{10}|G|$, then $G$ is solvable; this threshold is sharp, as $\mathcal{A}_5$ has an automorphism cycle of length exactly $\frac{1}{10}|\mathcal{A}_5|$.
• For periodic affine maps, solvability is implied if the cycle length exceeds $\frac{1}{4}|G|$, and this bound is also sharp, as $\mathcal{A}_5$ achieves a cycle of length $\frac{1}{4}|\mathcal{A}_5|$.
• For automorphisms, if the cycle length is at least $|G|^{1/3 + \epsilon}$, then $|\operatorname{Rad}(G)|$ grows faster than any $|G|^{1 - \frac{3}{2}\epsilon + \xi}$ for $\xi > 0$, implying $|\operatorname{Rad}(G)| \to \infty$ as $|G| \to \infty$.
• For periodic affine maps, if the cycle length is at least $|G|^{2/3 + \epsilon}$, then $|\operatorname{Rad}(G)|$ grows faster than any $|G|^{1 - 3\epsilon + \xi}$, so $|\operatorname{Rad}(G)| \to \infty$ as $|G| \to \infty$.
• There exists an infinite sequence of finite groups $G_n$ with trivial solvable radical ($|\operatorname{Rad}(G_n)| = 1$) such that each $G_n$ has an automorphism cycle of length greater than $|G_n|^{1/3}$ and a periodic affine map cycle of length greater than $|G_n|^{2/3}$, showing the thresholds are tight.
• The index $[G: \operatorname{Rad}(G)]$ is bounded above by $\rho^{E_1}$ for automorphisms and $\rho^{E_2}$ for periodic affine maps, where $E_1 \approx -1.778$ and $E_2 \approx -5.907$, showing that such groups are 'close to solvable'.