Khukhro, Evgeny and Moens, Wolfgang (2022) Fitting height of finite groups admitting a fixedpointfree automorphism satisfying an additional polynomial identity. Journal of Algebra, 608 . pp. 755773. ISSN 00218693
Full content URL: https://doi.org/10.1016/j.jalgebra.2022.07.006
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Item Type:  Article 

Item Status:  Live Archive 
Abstract
Let f(x) be a nonzero 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 fixedpointfree 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 nonzero 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 wellknown 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, fixedpointfree automorphism, Fitting height, HallHigman 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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