Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures

Brecht Verbeken

paper

Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures

arXiv:2608.03254

Abstract

Let $r > 1$ with $g cd (r, 30) = 1$ . We construct a finite group $G_{r} = (C_{5} \times C_{3} \times C_{r}) ⋊ W$ , where $∣ W ∣ = 32$ and $G_{r}^{'} ≅ C_{60 r}$ , together with an augmentation-preserving automorphism $α_{r} \in Aut (Z G_{r})$ having no Zassenhaus factorization. The image $Y_{r} = α_{r} (G_{r})$ is a normalized group basis which is not rationally conjugate to $G_{r}$ , although every element of $Y_{r}$ is individually rationally conjugate to an element of $G_{r}$ . Consequently, (ZC2) and (ZC3) fail for finite cyclic-by-abelian groups, resolving a problem of Margolis and del Río. The construction extends Hertweck's example uniformly: both the class-preserving obstruction and the integral gluing are independent of the order of the auxiliary factor. The smallest admissible member of this family has order $3360$ and derived subgroup $C_{420}$ .

17 pages; no figures