The Fitting height of finite groups with a fixed-point-free automorphism satisfying an identity

2021/10/18 by Wolfgang Alexander Moens, Moens, Wolfgang Alexander
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2110.09029

openalex publication_date 2021/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial f(x) = a₀ + a₁ x + ⋯ + a_d x^d ∈ ℤ[x] , there is an integer k > 0 with the following property. Let G be a finite (solvable) group with a fixed-point-free automorphism α satisfying gcd(|G|,k)= 1 and \ g^a₀ ⋅ α(g)^a₁ ⋅ α²(g)^a₂ ⋯ α^d(g)^a_d | g ∈ G \ = \1\. Then the Fitting height of G is at most the number of irreducible factors of f(x). We confirm the conjecture for a large family of polynomials with explicit constants k.

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