| Abstract: | Kronecker studied a class of groups 〈p, p - 1, r〉, whose commutator subgroups are prime cyclic of order p, and whose commutator quotient groups are cyclic of order p - 1. These are now commonly called the K-metacyclic groups. It follows from the classical work of Maschke that none of the K-metacyclic groups except 〈3, 2, 2〉 has a planar Cayley graph. It is proved here that only for p = 5 and p = 7 is a K-metacyclic group 〈p, p - 1, r〉 toroidal. To achieve this result, this paper develops a methodology for using Proulx's classification of toroidal groups by presentation to determine whether an explicitly given group is toroidal. |
