Premium
A determination of the toroidal k ‐metacyclic groups
Author(s) -
Gross Jonathan L.,
Lomonaco Samuel J.
Publication year - 1980
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190040205
Subject(s) - mathematics , combinatorics , quotient , toroid , prime (order theory) , kronecker delta , cyclic group , cayley graph , order (exchange) , commutator , graph , pure mathematics , algebra over a field , abelian group , physics , plasma , lie conformal algebra , finance , quantum mechanics , economics
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.