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

DOI10.1016/J.JALGEBRA.2022.07.006arXiv2201.08607MaRDI QIDQ6388821

Wolfgang Alexander Moens, Evgenii I. Khukhro

Publication date: 21 January 2022

Abstract: Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $varphi$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $varphi$ 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 $varphi$ 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 $operatornamedeg(f(x))$. We also prove that if $f(x)$ is any non-zero polynomial and $G$ is a $sigm{a}^{\prime }$-group for a finite set of primes $sigma=sigma(f(x))$ depending only on $f(x)$, then the Fitting height of $G$ is bounded in terms of the number $operatornameirr(f(x))$ 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 $alpha(|varphi|)$ of $langlevarphiangle$ when $operatornamedeg(f(x))$ or $operatornameirr(f(x))$ is small in comparison with $alpha(|varphi|)$.

Mathematics Subject Classification ID

Finite solvable groups, theory of formations, Schunck classes, Fitting classes, (pi)-length, ranks (20D10) Automorphisms of abstract finite groups (20D45)

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