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arXiv 1707.03090math.COmath.GR
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Haar图全为Cayley图的群

On groups all of whose Haar graphs are Cayley graphs

Yan-Quan Feng, Istvan Kovacs, Da-Wei Yang

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17 pages
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摘要：

一个群$H$的Cayley图是一个有限简单图$Γ$，使得${\rm Aut}(Γ)$包含一个同构于$H$的子群正则地作用在$V(Γ)$上，而$H$的Haar图是一个有限简单二部图$Σ$，使得${\rm Aut}(Σ)$包含一个同构于$H$的子群半正则地作用在$V(Σ)$上，并且$H$ -轨道等于$Σ$的二部集。Cayley图是Haar图，当它是二部图时，但Haar图是Cayley图的简单条件是未知的。本文证明了群$D_6, \, D_8, \, D_{10}$和$Q_8$是唯一的Haar图都是Cayley图的有限内阿贝尔群（如果一个群是非阿贝尔群，但它的所有真子群都是阿贝尔群，则称之为内阿贝尔群）.作为应用，还证明了每一个不可解群都有一个Haar图不是Cayley图。

英文摘要：

A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.

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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