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

2015/01/28 by Alexander Bors, Bors, Alexander · 1 citation
Chemistry · Mathematics · #20D25 #20D45 #20E22 #20G40 #37P99 #FOS: Mathematics #Ferrocene Chemistry and Applications #Finite Group Theory Research #Graph theory and applications #Group Theory (math.GR) #Primary: 20B25 #Secondary: 20D05

paper · pdf · doi:10.48550/arxiv.1501.07172

openalex publication_date 2015/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the following: For any ρ∈(0,1), if a finite group G has an automorphism with a cycle of length at least ρ⋅|G|, then the index of the solvable radical 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>(1)/(10). Furthermore, considering, for exponents e∈(0,1), 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 |Rad(G)|→∞ for |G|→∞ if and only if e>(1)/(3). We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.

Citations

• Classification of finite group automorphisms with a large cycle

Cited by

• A bound on element orders in the holomorph of a finite group

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