On infinite groups generated by two quaternions

Abstract

Let $x$, $y$ be two integral quaternions of norm $p$ and $l$, respectively, where $p$, $l$ are distinct odd prime numbers. We investigate the structure of $<x,y>$, the multiplicative group generated by $x$ and $y$. Under a certain condition which excludes $<x,y>$ from being free or abelian, we show for example that $<x,y>$, its center, commutator subgroup and abelianization are finitely presented infinite groups. We give many examples where our condition is satisfied and compute as an illustration a finite presentation of the group $<1+j+k, 1+2j>$ having these two generators and seven relations. In a second part, we study the basic question whether there exist commuting quaternions $x$ and $y$ for fixed $p$, $l$, using results on prime numbers of the form $r^2 + m s^2$ and a simple invariant for commutativity.