arXiv 1710.044462017-10-12math.GRmath.CO

character和不变但不是Cayley同构的非交换有限群

Non-abelian finite groups whose character sums are invariant but are not Cayley isomorphism

A. Abdollahi, M. Zallaghi

摘要：

设$G$是一个群，$S$是$G\setminus \{1\}$的逆闭子集. Cayley图$Cay(G,S)$是指其顶点集是$G$的元素集并且两个顶点$x$和$y$相邻的图，如果$x^{-1}y\in S$。一个群$G$称为CI-群，如果对于$G\setminus \{1\}$的逆闭子集$S$和$T$有$Cay(G,S)\cong Cay(G,T)$，则对于$G$的自同构$α$有$S^α=T$.一个有限群$G$称为BI-群，如果对于$G\setminus \{1\}$的某些逆闭子集$S$和$T$有$Cay(G,S)\cong Cay(G,T)$，则对于所有正整数$ν$有$M_ν^S=M_ν^T$，其中$M_ν^S$表示集合$\big\{\sum_{s\in S}χ(s) | χ(1)=ν, χ\text{ is a complex irreducible character of } G \big\}$。László Babai [ \textit{J. Combin. Theory Ser. B}，{ \bf 27}（1979）180-189]问到是否每个有限群都是BI-群;有限非BI-群的各种例子在[ \textit{Comm. Algebra}，{ \bf 43}（12）（2015）5159-5167]中给出。在后一篇文章中指出，每个有限CI-群是BI-群，所有交换有限群都是BI-群。然而，已知存在有限阿贝尔非CI-群。有限非交换BI-群（非CI-群）的存在性是本文研究的主要问题。我们找到了两个非交换的BI-群，它们的阶为$20$和$42$，但不是CI-群。我们还列出了$30$之前的所有订单的BI-组。

英文摘要：

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^α=T$ for some automorphism $α$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_ν^S=M_ν^T$ for all positive integers $ν$, where $M_ν^S$ denotes the set $\big\{\sum_{s\in S}χ(s) | χ(1)=ν, χ\text{ is a complex irreducible character of } G \big\}$. It was asked by László Babai [\textit{J. Combin. Theory Ser. B}, {\bf 27} (1979) 180-189] if every finite group is a BI-group; various examples of finite non BI-groups are presented in [\textit{Comm. Algebra}, {\bf 43} (12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However it is known that there are finite abelian non CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders $20$ and $42$ which are not CI-groups. We also list all BI-groups of orders up to $30$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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