A Cayley graph of a group H is a finite simple graph \(\Gamma \) such that \(\mathrm{Aut}(\Gamma )\) contains a subgroup isomorphic to H acting regularly on \(V(\Gamma ),\) while a Haar graph of H is a finite simple bipartite graph \(\Sigma \) such that \(\mathrm{Aut}(\Sigma )\) contains a subgroup isomorphic to H acting semiregularly on \(V(\Sigma )\) and the H-orbits are equal to the bipartite sets of \(\Sigma \). 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 \(D_6, \, D_8, \, D_{10}\) 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.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph.

中文翻译：

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在所有其Haar图为Cayley图的组上

H组的Cayley图是有限的简单图\（\ Gamma \），因此\（\ mathrm {Aut}（\ Gamma）\）包含与H规则同构地作用于\（V（\ Gamma）的子集， \），而H的Haar图是有限的简单二部图\（\ Sigma \），因此\（\ mathrm {Aut}（\ Sigma）\）包含与H同构的半同构作用于\ {V（\ Sigma ）\）和H轨道等于\（\ Sigma \）的二分集。Cayley图恰好是二分时才是Haar图，但是没有简单的条件可让Haar图成为Cayley图。在本文中，我们证明\（D_6，\，D_8，\，D_ {10} \）和\（Q_8 \）是仅有的有限内部阿贝尔群，它们的Haar图都是Cayley图。（如果一个组是非阿贝尔的，则称为内部阿贝尔，但其所有适当的子组都是阿贝尔的。）然后将使用此结果来推导每个不可解的组都承认一个非凯利Haar图。