Deficiency zero groups involving Fibonacci and Lucas numbers

Abstract

SynopsisTwo closely related classes X(n) and Y(n) of two generator two relation groups are studied. The group presentations arise from an investigation of a Fibonacci type group of order 1512. The Reidemeister-Schreier algorithm is used to show that the groups X(n) are finite and not metabelian. The orders of these groups are determined and shown to be divisible by powers of Fibonacci numbers or by powers of Lucas numbers. In addition these groups add to the relatively few examples of non-metabelian two generator two relation groups whose orders are known precisely.