On Prime Quaternions, Hurwitz Relations, and a~New Operation of Group Extension

S. I. Adian ; F. Grunevald ; J. Mennicke

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 7-22

Résumé

We study the Hurwitz relations that occur in the multiplicative group of Hamilton quaternions with rational coefficients. These relations arise for pairs of primary prime quaternions with prime norms $p$ and $q$. There are two permutation groups associated to the Hurwitz relations. We prove that these permutation groups are isomorphic to the groups $PSL(2,q)$, $PGL(2,q)$, $PSL(2,p)$, or $PGL(2,p)$. We also introduce a new extension operation for groups based on Hurwitz-type relations. The extension of a given finitely presented group $G$ uses a system of the so-called semistable letters, which are a generalization of the notion of stable letters introduced earlier by P. S. Novikov. The extensio $H$ of a given group $G$ is obtained by adding new generators and relations that satisfy the so-called normality condition. The extended group has a decidable word problem and a decidable conjugacy problem if the same problems are decidable for the given basic group.