Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Khukhro, Evgeny and Moens, Wolfgang (2022) Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity. Journal of Algebra, 608 . pp. 755-773. ISSN 0021-8693

Full content URL: https://doi.org/10.1016/j.jalgebra.2022.07.006

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Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity
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Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity
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Abstract

Let f(x) be a non-zero polynomial with integer coefficients. An automor phism 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 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 0 -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 different 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 hϕi when deg f(x) or irr(f(x)) is small in comparison with α(|ϕ|).

Keywords:finite group, fixed-point-free automorphism, Fitting height, Hall--Higman type theorems
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:50075
Deposited On:19 Jul 2022 13:21

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