Classification of finite group automorphisms with a large cycle

2014/10/08 by Alexander Bors, Bors, Alexander · 1 citation
Computer Science · Engineering · Mathematics · #12E20 #15A21 #20D45 #20F05 #20G40 #20K01 #20K30 #37P99 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary: 20B25 #Secondary: 11A07 #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1410.2284

openalex publication_date 2014/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)≥(1)/(2).

Cited by

• Cycle lengths in finite groups and the size of the solvable radical

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• Finite groups with an automorphism of large order
• Finite groups with an automorphism cubing a large fraction of elements
• Finite groups with a large automorphism orbit
• Computation of orders and cycle lengths of automorphisms of finite solvable groups
• On finite groups where the order of every automorphism is a cycle length