Absolutely split metacyclic groups and weak metacirculants

作者：Cui, Li; Zhou, Jin-Xin^*

来源：Journal of Algebra and Its Applications, 2019, 18(6): 1950117.

DOI：10.1142/S0219498819501172

• 2-Group
• absolutely split metacyclic group
• metacirculant

Let m, n, r be positive integers, and let G = < a > : < b > congruent to C-n : C-m be a split metacyclic group such that b(-1)ab = a(r). We say that C is absolutely split with respect to < a > provided that for any x is an element of G, if < x > boolean AND < a > = 1, then there exists y is an element of G such that x is an element of < 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 [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225-243]. 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 class of graphs, Marusic and Sparl in 2008 introduced 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.

• 出版日期2019-6
• 单位北京交通大学