On infinite groups generated by two quaternions
| dc.creator | Rattaggi, Diego | |
| dc.date | 2005-02-24 | |
| dc.date | 2006-10-06 | |
| dc.date.accessioned | 2026-07-07T06:39:28Z | |
| dc.date.available | 2026-07-07T06:39:28Z | |
| dc.description | 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. | |
| dc.description | 31 pages. Completely revised version, several new results and simplified proofs | |
| dc.identifier | https://arxiv.org/abs/math/0502512 | |
| dc.identifier | http://arxiv.org/abs/math/0502512 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/101092 | |
| dc.subject | Group Theory | |
| dc.subject | Rings and Algebras | |
| dc.title | On infinite groups generated by two quaternions | |
| dc.type | text |