Some families of finite groups having two generators and two relations

Abstract

A class of groups F a,b,c having two generators and two relations is studied. This class is of interest for two reasons. It is relevant to the study of trivalent O-symmetric graphs and also adds infinite families to the known class of finite groups with two-generator two-relation presentations. Adding a third relation to the presentation for F a’b’c gives a homomorphic image H a’b’c . In this paper the orders of the groups H a’b’c are determined, also the orders of certain subclasses of the groups F a’b’c . Finally a conjecture is given concerning the normal subgroup G such that H a , b , c ≅ F a , b , c / G .