On oriented \(m\)-semiregular representations of finite groups (Q6642503)

Let \(G\) be a finite group. A group \(G\) \textit{admits an ORR} (\textit{oriented regular representation}) if there exists a Cayley digraph \(\Gamma=\mathrm{Cay}(G,R)\) of \(G\) such that \(G\cong\textrm{Aut}(\Gamma)\) for some subset \(R\subseteq G\backslash\{1\}\) with \(R\cap R^{-1}=\emptyset\). \N\NA permutation group \(G\) on a set \(\Omega\) is called \textit{semiregular on} \(\Omega\) if \(G_{\omega} = 1\) for every \(\omega\in\Omega\) (\(G_{\omega}\) -- the subgroup of \(G\) fixing \(\omega\)) and \textit{regular} if it is semiregular and transitive. \N\NFor a positive integer \(m\), an \textit{\(m\)-Cayley digraph of \(G\)} is defined as a digraph which has a semiregular group of automorphisms isomorphic to \(G\) with \(m\) orbits on its vertex set. A group \(G\) \textit{admits an OmSR} (\textit{oriented \(m\)-semiregular representation}) if there exists an oriented \(m\)-Cayley digraph \(\Gamma\) of \(G\) satisfying \(G\cong \Aut(\Gamma)\). If \(\Gamma\) is regular, then it is called a \textit{regular oriented \(m\)-semiregular representation of \(G\)} (\textit{regular OmSR of} \(G\)). For each positive integer \(m\), some criteria for a finite group to admit a regular O\(m\)SR or an O\(m\)SR is deduced. For example, \(G\) admits an O2SR unless \(G\) is isomorphic to either \(\mathbb{Z}_2^2\) or \(\mathbb{Z}_2^3\) and \(G\) admits a regular O2SR unless \(G\) is isomorphic to \(\mathbb{Z}_1\), \(\mathbb{Z}_2\), \(\mathbb{Z}_2^2\), \(\mathbb{Z}_2^3\) or \(\mathbb{Z}_2^4\).