The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial $r(t) = a_0 + a_1 \cdot t + \cdots + a_d \cdot t^d \in \mathbb{Z}[t]$ that is irreducible over $\mathbb{Q}$, there exist (explicit) invariants $a,b,c \in \mathbb{N}$ with the following property. Consider a finite group with a fix-point-free automorphism ${\alpha}:{G}\longrightarrow{G}$ and suppose that for all $x \in G$ we have the equality $x^{a_0} \cdot \alpha(x^{a_1}) \cdot \alpha^2(x^{a_2})\cdots \alpha^d(x^{a_d}) = 1_G.$ Then $G$ is solvable and of the form $A \cdot (B \rtimes (C \times D))$, where $A$ is an $a$-group, $B$ is a $b$-group, $C$ is a nilpotent $c$-group, and $D$ is a nilpotent group of class at most $d^{2^d}$. Here, a group $H$ is said to be an $a$-group (resp. $b$-group or $c$-group) if the order of every $h \in H$ divides some natural power of $a$ (resp. $b$ or $c$).