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 \(c\)).