论文标题
在$ \ bci $ -groups和$ \ ci $ -groups上
On $\BCI$-groups and $\CI$-groups
论文作者
论文摘要
令$ g $为有限的组，$ s $为$g。$ a bi-cayley图$ \ bcay（g，s）$是一个简单且无方向的图形，带有vertex-set $ g \ times \ times \ times \ {1,2 \} $ {1,2 \} $ and Edge-set $ \ \ \ \ \ {（g，1） s \} $。 Bi-cayley图$ \ bcay（g，s）$被称为$ \ bci $ -graph，如果任何bi-cayley图$ \ bcay（g，t）$，每当$ \ bcay（g，s）\ cong \ cong \ bcay（g，g，t） $ g $如果$ g $的每个双 - 与$ g $的图形都是$ \ bci $ -graph，则称为$ \ bci $ -group。在本文中，我们表明，每个$ \ bci $ -group都是$ \ ci $ -group，它对Arezoomand和Taeri在\ cite {arezoomand1}中提出的猜想提供了积极的答案。另外，我们证明了没有任何非亚伯利亚$ 4 $ - $ \ bci $ -simple Group。此外，所有$ \ bci $ - 订单$ 2p $，$ p $ a Prime的订单都被表征了。
Let $G$ be a finite group and $S$ be a subset of $G.$ A bi-Cayley graph $\BCay(G,S)$ is a simple and an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. A bi-Cayley graph $\BCay(G,S)$ is called a $\BCI$-graph if for any bi-Cayley graph $\BCay(G,T)$, whenever $\BCay(G,S)\cong\BCay(G,T)$ we have $T=gS^σ$ for some $g\in G$ and $σ\in\Aut(G).$ A group $G$ is called a $\BCI$-group if every bi-Cayley graph of $G$ is a $\BCI$-graph. In this paper, we showed that every $\BCI$-group is a $\CI$-group, which gives a positive answer to a conjecture proposed by Arezoomand and Taeri in \cite{arezoomand1}. Also we proved that there is no any non-Abelian $4$-$\BCI$-simple group. In addition all $\BCI$-groups of order $2p$, $p$ a prime, are characterized.
