Absolutely split metacyclic groups and weak metacirculants

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Let m,n,r be positive integers, and let G= a: b Zₙ: Zₘ be a split metacyclic group such that b⁻¹ab=aʳ. We say that G is absolutely split with respect to a provided that for any x∈ G, if x∩ a=1, then there exists y∈ G such that x∈ y and G= a: y. 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 si c and Sparl in 2008 posed the so called weak metacirculants. A graph is called a 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.