Generalized quaternion groups of order 4p are $4$-BCI-groups

Abstract

For a finite group G and a subset S⊆ G, the bi-Cayley graph BCay(G,S) has vertex set G×{0,1} and edges {(x,0),(sx,1)} for x∈ G and s∈ S. The group G is called an m-BCI-group if, for all subsets S,T⊆ G with |S|≤ m, the isomorphism BCay(G,S)≅BCay(G,T) implies that T=gS^α for some g∈ G and α∈Aut G. We prove that, for every prime p, the generalized quaternion group Q_4p is a $4$-BCI-group. For p≥5, the proof uses regular coverings and voltage assignments: we pass to quotient graphs modulo the Sylow p-subgroup and lift isomorphisms between them.

Realizable Cohomology Classes of Dihedral and Generalized Quaternion Groups
Sutirtha Datta, Jyotirmoy Ganguly, Rohit Joshi
arXiv23 Oct 2026PreprintCC BY-NC-ND 4.0Full textAlgebra & number theory Geometry & topologyarXiv
On isomorphisms of tetravalent Cayley graphs over generalized quaternion groups
Lele Jin, Li Ma, Wei Meng
arXiv28 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
Amir Hossein Nokhodkar
arXiv4 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Bitwise Triangular Coordinates for Central Products of Quaternion Groups: Floretion Base Vectors, Digitwise S3-Actions, and Centralizer Tiles
Creighton Dement
arXiv16 Sep 2026PreprintarXiv licenceFull textGeneral mathematicsarXiv
Order-Specified Quaternion CP Decomposition: Models, Algorithms, and Applications
Jifei Miao, Niansheng Tang, Michael K. Ng +2
arXiv21 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv

Cite

BibTeX RIS

BibTeX

@misc{xie2026generalized,
  title = {{Generalized quaternion groups of order $4p$ are $4$-BCI-groups}},
  author = {Jin-Hua Xie},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25378},
  eprint = {2610.25378},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25378},
}

RIS

TY  - UNPB
TI  - Generalized quaternion groups of order 4p are $4$-BCI-groups
AU  - Jin-Hua Xie
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25378
UR  - https://arxiv.org/abs/2610.25378
AB  - For a finite group G and a subset S⊆ G, the bi-Cayley graph BCay(G,S) has vertex set G×{0,1} and edges {(x,0),(sx,1)} for x∈ G and s∈ S. The group G is called an m-BCI-group if, for all subsets S,T⊆ G with |S|≤ m, the isomorphism BCay(G,S)≅BCay(G,T) implies that T=gS^α for some g∈ G and α∈Aut G. We prove that, for every prime p, the generalized quaternion group Q_4p is a $4$-BCI-group. For p≥5, the proof uses regular coverings and voltage assignments: we pass to quotient graphs modulo the Sylow p-subgroup and lift isomorphisms between them.
ER  -

Abstract

Related

Cite