Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

Abstract

It is known that a finite group with an automorphism $\phi$ of coprime order has a soluble radical of $(|\phi |,|C_G(\phi )|)$-bounded Fitting height and index. We extend this classic result as follows. Let $f(x)=a_0+a_1\cdot x+\cdots +a_d\cdot x^d\in \mathbb{Z}[x]$ be a primitive polynomial and let $G$ be a finite group with an automorphism $\phi$ of coprime order satisfying $g^{a_0}\cdot \phi (g{)}^{a_1}\cdots {\phi }^d(g{)}^{a_d}=1$, for all $g\in G$. Then the soluble radical of $G$ has $(d,|C_G(\phi )|)$-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree $d$.