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

Abstract

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|).