The graphical regular representations of finite metacyclic \(p\)-groups
Let a group \(G\) and a subset \(S\) of \(G\) that does not contain the identity element of \(G\) be given. Let \(\text{Cay}(G,S)\) be a Cayley graph of group \(G\) with vertex set \(G\) and edge set \(\{\{x,sx\}\mid x \in G\), \(s \in S\), \(S = S^{-1}\}.\) If the full automorphism group of \(\text{Cay}(G,S)\) is equal to \(G,\) then \(\text{Cay}(G,S)\) is called a graphical regular representation of \(G.\) Let \(\Aut(G,S)\) be the subgroup of the full automorphism group of \(G\) which consists of the elements \(\theta\) such that \(\theta(S) = S.\) The paper concerns a problem of Godsil: For a given class of groups \(G\), determine a condition under which \(\text{Cay}(G,S)\) is a graphical regular representation of \(G\) if and only if \(\Aut(G,S) = 1.\) Godsil's problem was solved (i) for \(p\)-groups which have no homomorphism onto \(Z_{p}\wr Z_{p},\) and for dihedral \(2\)-groups by \textit{C. D. Godsil} [Combinatorica 1, 243-256 (1981; Zbl 0489.05028)]; (ii) for cubic Cayley graphs of alternating groups of degree at least 19 and for certain cubic Cayley graphs of 2-power order by \textit{C. D. Godsil} [Eur. J. Comb. 4, 25-32 (1983; Zbl 0507.05038)]; and (iii) for arbitrary cubic Cayley graphs of 2-power order by \textit{C. H. Li} [J. Comb. Theory, Ser. B 72, 140-142 (1998; Zbl 0897.05046)]. The main result of the paper under review reads: If \(p\) is a prime and \(G\) a finite metacyclic \(p\)-group, then the Cayley graph \(\text{Cay}(G,S)\) is a graphical regular representation of \(G\) if and only if \(\Aut(G,S) = 1\) (Theorem 1.1). Let \(A\) be the full automorphism group \(\Aut \Gamma\) of a Cayley graph and let \(A_1\) be the stabilizer in \(A\) of the identity element \(1\) of \(G.\) The authors also show that if \(G\) is a nonabelian metacyclic \(2\)-group, and \(\Gamma = \text{Cay}(G,S)\) is connected such that \(A_1\) is of odd order, then \(G\) is normal in \(\Aut \Gamma\) (Theorem 4.3).
full automorphism group of a Cayley graph
scientific article; zbMATH DE number 1538574
scientific article