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 Tr. Mat. Inst. Steklova, 2003, Volume 242, Pages 7–22 (Mi tm402)

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

S. I. Adiana, F. Grunevaldb, J. Mennickec

a Steklov Mathematical Institute, Russian Academy of Sciences
b Heinrich-Heine-Universität Düsseldorf
c Bielefeld University

Abstract: 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.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 242, 3–17

Bibliographic databases:

Document Type: Article
UDC: 512
Received in September 2002

Citation: S. I. Adian, F. Grunevald, J. Mennicke, “On Prime Quaternions, Hurwitz Relations, and a New Operation of Group Extension”, Mathematical logic and algebra, Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov, Tr. Mat. Inst. Steklova, 242, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 7–22; Proc. Steklov Inst. Math., 242 (2003), 3–17

Citation in format AMSBIB
\Bibitem{AdiGruMen03} \by S.~I.~Adian, F.~Grunevald, J.~Mennicke \paper On Prime Quaternions, Hurwitz Relations, and a~New Operation of Group Extension \inbook Mathematical logic and algebra \bookinfo Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 242 \pages 7--22 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm402} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054482} \zmath{https://zbmath.org/?q=an:1113.20030} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 242 \pages 3--17 

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