Geodesic
On Prime Quaternions, Hurwitz Relations, and a New Operation of Group Extension
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 7-22 
S. I. Adian; F. Grunevald; J. Mennicke. On Prime Quaternions, Hurwitz Relations, and a New Operation of Group Extension. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical logic and algebra, Tome 242 (2003), pp. 7-22. http://geodesic.mathdoc.fr/item/TM_2003_242_a1/
@article{TM_2003_242_a1,
     author = {S. I. Adian and F. Grunevald and J. Mennicke},
     title = {On {Prime} {Quaternions,} {Hurwitz} {Relations,} and {a~New} {Operation} of {Group} {Extension}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {7--22},
     year = {2003},
     volume = {242},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_242_a1/}
}
TY  - JOUR
AU  - S. I. Adian
AU  - F. Grunevald
AU  - J. Mennicke
TI  - On Prime Quaternions, Hurwitz Relations, and a New Operation of Group Extension
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2003
SP  - 7
EP  - 22
VL  - 242
UR  - http://geodesic.mathdoc.fr/item/TM_2003_242_a1/
LA  - ru
ID  - TM_2003_242_a1
ER  - 
%0 Journal Article
%A S. I. Adian
%A F. Grunevald
%A J. Mennicke
%T On Prime Quaternions, Hurwitz Relations, and a New Operation of Group Extension
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2003
%P 7-22
%V 242
%U http://geodesic.mathdoc.fr/item/TM_2003_242_a1/
%G ru
%F TM_2003_242_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] Hurwitz A., “Ueber die Zahlentheorie der Quaternionen”, Nachr. Ges. Wiss. Göttingen. Math.-phys. Kl., 1896, 313–340; Mathematische Werke, Bd. 2, Birkhäuser, Basel, Stuttgart, 1963, 303–330

[2] Adian S. I., Grunewald F., Lysionok I. G., Mennicke J., “On embeddings of $SL(2,\mathbb{Z})$ into quaternion groups”, Math. Ztschr., 238 (2001), 389–399 | DOI | MR | Zbl

[3] Grunewald F., Segal D., “Decision problems concerning $S$-arithmetic groups”, J. Symb. Logic, 50:3 (1985), 743–772 | DOI | MR | Zbl

[4] Novikov P. S., Ob algoritmicheskoi nerazreshimosti problemy tozhdestva teorii grupp, Tr. MIAN, 44, AN SSSR, M., 1955 | MR | Zbl

[5] Adyan S. I., Durnev V. G., “Algoritmicheskie problemy dlya grupp i polugrupp”, UMN, 55:2 (2000), 3–94 | MR | Zbl

[6] Dickson L. E., Modern elementary theory of numbers, Univ. Chicago Press, Chicago, 1939

[7] Huppert B., Endliche Gruppen, I, Springer, Berlin, 1967 | MR | Zbl

[8] Reiner I., Maximal orders, London Math. Soc. Monogr., 5, Acad. Press, London, 1975 | MR | Zbl