Some families of finite groups having two generators and two relations

Abstract

Abstract A class of groups Fa,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 Fa’b’c gives a homomorphic image Ha’b’c. In this paper the orders of the groups Ha’b’c are determined, also the orders of certain subclasses of the groups Fa’b’c. Finally a conjecture is given concerning the normal subgroup G such that Ha,b,c≅Fa,b,c/G.