Samuel J. Lomonaco, Jr.

**
ABSTRACT.**

**
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.
**