Year: 2026
Type: preprint
Abstract: For a finite group $G$ and a subset $S\subseteq G$, the bi-Cayley graph $\operatorname{BCay}(G,S)$ has vertex set $G\times\{0,1\}$ and edges $\{(x,0),(sx,1)\}$ for $x\in G$ and $s\in S$. The group $G$ is called an $m$-BCI-group if, for all subsets $S,T\subseteq G$ with $|S|\leq m$, the isomorphism $\operatorname{BCay}(G,S)\cong\operatorname{BCay}(G,T)$ implies that $T=gS^α$ for some $g\in G$ and $α\in\operatorname{Aut} G$. We prove that, for every prime $p$, the generalized quaternion group $Q_{4p}$ is a $4$-BCI-group. For $p\geq5$, the proof uses regular coverings and voltage assignments: we pass to quotient graphs modulo the Sylow $p$-subgroup and lift isomorphisms between them.
Source: arXiv (Cornell University)
Author: Xie, Jin-Hua
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223005836