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

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.