Year: 2026
Type: preprint
Abstract: Let $r>1$ with $\gcd(r,30)=1$. We construct a finite group $G_r=(C_5\times C_3\times C_r)\rtimes W$, where $|W|=32$ and $G_r'\cong C_{60r}$, together with an augmentation-preserving automorphism $α_r\in\operatorname{Aut}(\mathbb{Z}G_r)$ 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 $C_{420}$.
Source: arXiv (Cornell University)
Author: Verbeken, Brecht F.
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172523604