A classification of nilpotent $3$-BCI groups

Document Type: Research Paper

Authors

1 National Autonomous University of Mexico

2 University of Primorska

Abstract

‎‎Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex‎ ‎set is $G \times \{0,1\}$ and edge set is‎ ‎$\{ \{(x,0),(s x,1)\}‎ : ‎x \in G‎, ‎s\in S \}$‎. ‎A bi-Cayley graph $\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph‎ ‎$\bcay(G,T),$ $\bcay(G,S) \cong \bcay(G,T)$ implies that $T = g S^\alpha$ for some $g \in G$ and $\alpha \in \aut(G)$‎. ‎A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs‎. ‎It was proved by Jin and Liu that‎, ‎if $G$ is a $3$-BCI-group‎, ‎then its Sylow $2$-subgroup is cyclic‎, ‎or elementary abelian‎, ‎or $\Q$ [European J‎. ‎Combin‎. ‎31 (2010)‎ ‎1257--1264]‎, ‎and that a Sylow $p$-subgroup‎, ‎$p$ is an odd prime‎, ‎is homocyclic [Util‎. ‎Math‎. ‎86 (2011) 313--320]‎. ‎In this paper we show that the converse also holds in the‎ ‎case when $G$ is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent $3$-BCI-groups‎.

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