On infinite groups generated by two quaternions

Details On infinite groups generated by two quaternions On infinite groups generated by two quaternions

2005-02-24

arxiv.org/abs/math/0502512v5

Mathematics

Group Theory

31 pages. Completely revised version, several new results and simplified proofs

Scientific paper

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 $$, the multiplicative group generated by $x$ and $y$. Under a certain condition which excludes $$ from being free or abelian, we show for example that $$, 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.

Affiliated with

Also associated with

No associations

Rating

On infinite groups generated by two quaternions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On infinite groups generated by two quaternions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On infinite groups generated by two quaternions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-235673