A determination of the toroidal k-metacyclic groups

✍ Scribed by Jonathan L. Gross; Samuel J. Lomonaco Jr.

• Publisher: John Wiley and Sons
• Year: 1980
• Tongue: English
• Weight: 387 KB
• Volume: 4
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.3190040205

✦ Synopsis

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.