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

Abstract

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 $\alpha :G⟶G$ 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 $A\cdot (B⋊(C\times D))$, 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 $c$).

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