Absolutely split metacyclic groups and weak metacirculants

Let $m,n,r$ be positive integers, and let $G=\langle a\rangle: \langle b\rangle \cong \mathbb{Z}_n: \mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {\em absolutely split with respect to $\langle a\rangle$} provided that for any $x\in G$, if $\langle x\rangle\cap\langle a\rangle=1$, then there exists $y\in G$ such that $x\in\langle y\rangle$ and $G=\langle a\rangle: \langle y\rangle$. In this paper, we give a sufficient and necessary condition for the group $G$ being absolutely split. This generalizes a result of Sanming Zhou and the second author in [arXiv: 1611.06264v1]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in $1982$ and have been a rich source of various topics since then. As a generalization of this classes of graphs, Maru\v si\v c and \v Sparl in 2008 posed the so called weak metacirculants. A graph is called a {\em weak metacirculant} if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of $2$-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.