2022/01/21 by E. I. Khukhro, Khukhro, E. I., W. A. Moens +1 · 1 citation
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2201.08607
openalex publication_date 2022/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f(x) be a non-zero polynomial with integer coefficients. An automorphism φ of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by φ on every characteristic elementary abelian section S of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism φ satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a σ'-group for a finite set of primes σ=σ(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|φ|) of ⟨φ⟩ when deg (f(x)) or irr(f(x)) is small in comparison with α(|φ|).