A Determination of the Toroidal K-Metacyclic Groups

Jonathan L. Gross
Samuel J. Lomonaco, Jr.


Kronecker studied a class of groups , 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 classical work of Maschke that none of the K-metacyclic groupps 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 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.