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A classification of nilpotent $3$-BCI groups | ||
| International Journal of Group Theory | ||
| مقاله 8، دوره 8، شماره 2، شهریور 2019، صفحه 11-24 اصل مقاله (244.45 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.100795.1404 | ||
| نویسندگان | ||
| Hiroki Koike1؛ Istvan Kovacs* 2 | ||
| 1National Autonomous University of Mexico | ||
| 2University of Primorska | ||
| چکیده | ||
| 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 $Q_8$ [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. | ||
| کلیدواژهها | ||
| bi-Cayley graph؛ BCI-group؛ graph isomorphism | ||
| مراجع | ||
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