On quasiabelian Cayley graphs and graphical doubly regular representations
A graph \(\Gamma\) is a graphical doubly regular representation (GDRR) of a group \(G\) if \(\Gamma\) is a Cayley graph of \(G\) and \(\text{Aut}(\Gamma)=L(G)R(G)\langle\zeta_G\rangle\). Here \(L(G)\) and \(R(G)\) are respectively the left and right regular representations of \(G\), and \(\zeta_G\) is the permutation \(g\leftrightarrow g^{-1}\). The main question underlying the numerous results in this article is, which finite groups admit a GDRR? The Cayley graph \(\text{Cay}(G,S)\), where \(1\notin S=S^{-1}\), is called quasiabelian if \(S\) is a normal set. Define \(G\bowtie G=(G\times G)/\{(g,g):g\in Z(G)\}\), and define the central closure of \(G\) to be \(G\bowtie G\cong G\) if \(G\) is an elementary abelian 2-group, and the semidirect product of \(G\bowtie G\) by \(\zeta_G\) where \(\zeta_G((g,h))\zeta_G^{-1}=(h,g)\) otherwise. It is shown that the automorphism group of a quasiabelian Cayley graph of \(G\neq\{1\}\) always contains an isomorphic copy of \(G\bowtie G\) and is a GDRR of \(G\) if and only if its automorphism group is isomorphic to \(G\bowtie G\). As a corollary the author is able to determine directly from a result by \textit{W. Imrich} and the reviewer [Period Math. Hung. 7, 243-258 (1976; Zbl 0335.05118)] that with eight small exceptions, every finite abelian group admits a GDRR. Among the groups not admitting a GDRR are the alternating groups \(A_n\) for \(n\geq 4\), nonabelian groups of order \(p^3\) for \(p\) prime, and every group of which every nontrivial conguacy class is a coset of its center. If there exists one ``main result'' it is the following. If a Cayley graph is a GDRR of two nonisomorphic groups, then both groups are nonabelian with nontrivial isomorphic centers, isomorphic central quotients, and isomorphic normal subgroup lattices. In addition to numerous examples and nonexamples, there is a characterization of disconnected GDRR's, a characterization of their lattice of systems of (blocks of) imprimitivity, a characterization of when the central closure is decomposable, a list of problems for further research, and a list of some 93 references pertinent to this material.
scientific article; zbMATH DE number 1743047