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Primitivity testing of finite nilpotent linear groups

Part of: Representation theory of groups Other groups of matrices

Published online by Cambridge University Press:  01 March 2011

Tobias Rossmann
Tobias Rossmann*
Affiliation:
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland (email: [email protected])

Abstract

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We describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups 20H20: Other matrix groups over fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 87 - 98
Copyright
Copyright © London Mathematical Society 2011

References

[1]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.CrossRefGoogle Scholar
[2]Celler, F., Leedham-Green, C. R., Murray, S. H., Niemeyer, A. C. and O’Brien, E. A., ‘Generating random elements of a finite group’, Comm. Algebra 23 (1995) no. 13, 4931–4948.CrossRefGoogle Scholar
[3]Cohen, H., Number theory. Tools and Diophantine equations, vol. 1, Graduate Texts in Mathematics 239 (Springer, New York, 2007).Google Scholar
[4]Detinko, A. S. and Flannery, D. L., ‘Computing in nilpotent matrix groups’, LMS J. Comput. Math. 9 (2006) 104–134 (electronic).CrossRefGoogle Scholar
[5]Fein, B., Gordon, B. and Smith, J. H., ‘On the representation of −1 as a sum of two squares in an algebraic number field’, J. Number Theory 3 (1971) 310–315.CrossRefGoogle Scholar
[6]Fieker, C., ‘Über relative Normgleichungen in algebraischen Zahlkörpern’. PhD Thesis, Technische Universität Berlin, 1997.Google Scholar
[7]Holt, D. F., Eick, B. and O’Brien, E. A., Handbook of computational group theory, Discrete Mathematics and its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2005).CrossRefGoogle Scholar
[8]Holt, D. F., Leedham-Green, C. R., O’Brien, E. A. and Rees, S., ‘Testing matrix groups for primitivity’, J. Algebra 184 (1996) no. 3, 795–817.CrossRefGoogle Scholar
[9]Huppert, B., Character theory of finite groups, De Gruyter Expositions in Mathematics 25 (Walter de Gruyter, Berlin, 1998).CrossRefGoogle Scholar
[10]Lam, T. Y., Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67 (American Mathematical Society, Providence, RI, 2005).Google Scholar
[11]Leedham-Green, C. R. and McKay, S., The structure of groups of prime power order, London Mathematical Society Monographs. New Series 27 (Oxford University Press, Oxford, 2002) Oxford Science Publications.CrossRefGoogle Scholar
[12]Lorenz, F., Algebra, vol. II, Universitext (Springer, New York, 2008) ; Fields with structure, algebras and advanced topics. Translated from the German by Silvio Levy.Google Scholar
[13]Neukirch, J., Algebraische Zahlentheorie (Springer, Berlin, 1992).CrossRefGoogle Scholar
[14]Roquette, P., ‘Realisierung von Darstellungen endlicher nilpotenter Gruppen’, Arch. Math. (Basel) 9 (1958) 241–250.CrossRefGoogle Scholar
[15]Rossmann, T., ‘Irreducibility testing of finite nilpotent linear groups’, J. Algebra 324 (2010) no. 5, 1114–1124.CrossRefGoogle Scholar
[16]Rossmann, T., ‘finn — computing with finite nilpotent linear groups, 0.5’ 2010, seehttp://www.maths.nuigalway.ie/∼tobias/finn.Google Scholar
[17]Simon, D., ‘Solving norm equations in relative number fields using S-units’, Math. Comp. 71 (2002) no. 239, 1287–1305 (electronic).CrossRefGoogle Scholar
[18]Suprunenko, D. A., Matrix groups (American Mathematical Society, Providence, RI, 1976) ; Translated from the Russian, translation edited by K. A. Hirsch, Translations of Mathematical Monographs 45.CrossRefGoogle Scholar
[19]Wehrfritz, B. A. F., Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Matematik und ihrer Grenzgebiete 76 (Springer, New York, 1973).Google Scholar