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PreprintCC BY 4.0arXiv

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

  • Brecht F. Verbeken
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Abstract

Let r>1 with gcd(r,30)=1. We construct a finite group Gr=(C5× C3× Cr)rtimes W, where |W|=32 and Gr'≅ C60r, together with an augmentation-preserving automorphism αr∈Aut(ℤGr) having no Zassenhaus factorization. The image Yr=αr(Gr) is a normalized group basis which is not rationally conjugate to Gr, although every element of Yr is individually rationally conjugate to an element of Gr. 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 C420.

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Cite

BibTeXRIS
BibTeX
@misc{verbeken2026cyclicbyabelian,
  title = {{Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures}},
  author = {Brecht F. Verbeken},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.03254},
  eprint = {2608.03254},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.03254},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures
AU  - Brecht F. Verbeken
PY  - 2026
DA  - 2026/08/04
PB  - arXiv
DO  - 10.48550/arxiv.2608.03254
UR  - https://arxiv.org/abs/2608.03254
AB  - Let r>1 with gcd(r,30)=1. We construct a finite group G_r=(C_5× C_3× C_r)rtimes W, where |W|=32 and G_r'≅ C_60r, together with an augmentation-preserving automorphism α_r∈Aut(ℤG_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.
ER  -