### Impacto

Corrales Rodrigáñez, Carmen and Jespers, Eric and Leal, Guilherme and Rio, Ángel del
(2004)
*Presentations of the unit group of an order in a non-split quaternion algebra.*
Advances in Mathematics, 186
(2).
pp. 498-524.
ISSN 0001-8708

PDF
Restringido a Repository staff only 644kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0001870803002585

## Abstract

We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra H(K) over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of H(Z[(1+root-7)/(2)]). As a consequence a presentation is discovered for the orthogonal group SO3(Z[(1+root-7)/(2)]). These results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non-commutative division algebra that is not a totally definite quaternion algebra.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Algorithms; Finite sets of generators; Unit groups; Orders; quaternion algebras; Presentations |

Subjects: | Sciences > Mathematics > Group Theory |

ID Code: | 14942 |

References: | [1] A.F. Beardon, The Geometry of Discrete Groups, springer, Berlin, 1983. [2] L. Bianchi, Sui gruppi de sostituzioni lineari con coeficienti appartenenti a corpi quadratici imaginari, Math. Ann. 40 (1892) 332–412. [3] J. Elstrodt, F. Grunewald, J. Mennicke, Groups Acting on Hyperbolic Space, Harmonic Analysis and Number Theory, Springer, Berlin, 1998. [4] B. Fein, B. Gordon, J.M. Smith, On the representation of 1 as a sum of two squares in an algebraic number field, J. Number Theory 3 (1971) 310–315. [5] B. Fine, The Algebraic Theory of the Bianchi Groups, Marcel Dekker, New York, 1989. [6] A.J. Hahn, O.T. O’Meara, The Classical Groups and K-Theory, Grundlehren der mathematischen Wissenschaften 291, Springer, Heidelberg, 1989. [7] E. Jespers, Units in integral group rings: a survey, Proceedings of the International Conference on Methods in Ring Theory, Trento, 1997. Lecture Notes in Pure and Applied Mathematics, Vol. 198,Marcel Dekker, New York, 1998, pp. 141–169. [8] E. Kleinert, Units in Skew Fields, Progress in Mathematics, 186, Birkha¨ user Verlag, Basel, 2000. [9] E. Kleinert, Units of classical orders: a survey, Enseign. Math. (2) 40 (3–4) (1994) 205–248. [10] H. Poincare´, Me´moire sur les groupes kleine´ es, Acta. Math. 3 (1883) 49–92. [11] R. Riley, Applications of a computer implementation of Poincare´ ’s theorem on fundamental polyhedra, Math. Comp. 40 (162) (1983) 607–632. |

Deposited On: | 20 Apr 2012 11:51 |

Last Modified: | 15 Jan 2016 14:56 |

Repository Staff Only: item control page