On Infinite Groups Generated by Two Quaternions

Let x, y be two integer quaternions of norm p and l, respectively, where p, l are distinct odd prime numbers. What can be said about 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 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 + ns and a simple invariant for commutativity.