Curran, S.J., Hamilton paths in Cayley digraphs of metacyclic groups, Discrete Mathematics 115 (1993) 133-139. We obtain a characterization of all Hamilton paths in the Cayley digraph of a metacyclic group G with generating set {x, y} where (yx-‘) a G. The abundance of these Hamilton paths allows us to show that Hamilton paths occur in groups of at least two.

We show that if the fundamental group π of an inde-composable orientable P D 3-complex is the fundamental group of a finite graph of finite groups then the vertex groups have periodic cohomology and the edge groups are metacyclic. If the vertex groups all have cohomological period dividing 4 at most one is not dihedral, the edge groups are Z/2Z, and the underlying graph is a tree. We also ask w...

This question is rather old [3]: Which groups having periodic cohomology can act freely on some homotopy sphere? The first nontrivial restriction was supplied by Milnor in [3]. Every element of order two must lie in the center. Swan [ô] showed that any group with periodic cohomology acts freely on a c.w. complex of the homotopy type of sphere. If the group has odd order then it is metacyclic. T...

The chiral hypermaps with at most four hyperfaces were classified in [A. Breda d’Azevedo and R. Nedela,Chiral hypermaps with few hyperfaces, Math. Slovaca, 53 (2003), n.2, 107–128]. It arises from this classification that all chiral hypermaps are “canonical metacyclic”, that is, the one-step rotation about a hypervertex, or about a hyperedge, or about a hyperface, generates a normal subgroup in...