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.
Koike, H., Kovacs, I. (2019). A classification of nilpotent $3$-BCI groups. International Journal of Group Theory, 8(2), 11-24. doi: 10.22108/ijgt.2017.100795.1404
MLA
Hiroki Koike; Istvan Kovacs. "A classification of nilpotent $3$-BCI groups". International Journal of Group Theory, 8, 2, 2019, 11-24. doi: 10.22108/ijgt.2017.100795.1404
HARVARD
Koike, H., Kovacs, I. (2019). 'A classification of nilpotent $3$-BCI groups', International Journal of Group Theory, 8(2), pp. 11-24. doi: 10.22108/ijgt.2017.100795.1404
VANCOUVER
Koike, H., Kovacs, I. A classification of nilpotent $3$-BCI groups. International Journal of Group Theory, 2019; 8(2): 11-24. doi: 10.22108/ijgt.2017.100795.1404
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