Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

2022/02/21 by Wolfgang Alexander Moens, Moens, Wolfgang Alexander
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2202.10087

openalex publication_date 2022/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

It is known that a finite group with an automorphism φ of coprime order has a soluble radical of (|φ|,|C_G(φ)|)-bounded Fitting height and index. We extend this classic result as follows. Let f(x) = a₀ + a₁ ⋅ x + ⋯ + a_d ⋅ x^d ∈ ℤ[x] be a primitive polynomial and let G be a finite group with an automorphism φ of coprime order satisfying g^a₀ ⋅ φ(g)^a₁ ⋯ φ^d(g)^a_d = 1 , for all g ∈ G. Then the soluble radical of G has (d,|C_G(φ)|)-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree d.

Related

• The Fitting height of finite groups with a fixed-point-free automorphism satisfying an identity
• On finite soluble groups with almost fixed-point-free automorphisms of non-coprime order
• Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity
• On the Fitting height of soluble groups admitting a coprime factorisation
• Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks