Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Torsion units in integral group rings of Janko simple groups


Authors: V. A. Bovdi, E. Jespers and A. B. Konovalov
Journal: Math. Comp. 80 (2011), 593-615
MSC (2010): Primary 16S34, 20C05; Secondary 20D08
DOI: https://doi.org/10.1090/S0025-5718-2010-02376-2
Published electronically: June 9, 2010
MathSciNet review: 2728996
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $ J_1$, $ J_2$ and $ J_3$ is the same as that of the normalized unit group of their respective integral group ring.


References [Enhancements On Off] (What's this?)

  • 1. V. A. Artamonov and A. A. Bovdi, Integral group rings: groups of invertible elements and classical $ K$-theory, Algebra. Topology. Geometry, Vol. 27 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 3-43, 232. MR 1039822 (91e:16028)
  • 2. V. Bovdi, A. Grishkov, and A. Konovalov, Kimmerle conjecture for the Held and O'Nan sporadic simple groups, Sci. Math. Jpn. 69 (2009), no. 3, 353-361. MR 2510100
  • 3. V. Bovdi and M. Hertweck, Zassenhaus conjecture for central extensions of $ {S}_{5}$, J. Group Theory 11 (2008), no. 1, 63-74. MR 2381018
  • 4. V. Bovdi and A. Konovalov, Integral group ring of the first Mathieu simple group, Groups St. Andrews 2005. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 339, Cambridge Univ. Press, Cambridge, 2007, pp. 237-245. MR 2328163
  • 5. -, Integral group ring of the McLaughlin simple group, Algebra Discrete Math. (2007), no. 2, 43-53. MR 2364062
  • 6. -, Integral group ring of the Mathieu simple group $ M\sb{23}$, Comm. Algebra 36 (2008), no. 7, 2670-2680. MR 2422512
  • 7. -, Torsion units in integral group ring of Higman-Sims simple group, Studia Sci. Math. Hungar. (to appear, 2009).
  • 8. V. Bovdi, A. Konovalov, and S. Linton, Torsion units in integral group ring of the Mathieu simple group $ {M}_{22}$, LMS J. Comput. Math. 11 (2008), 28-39. MR 2379938
  • 9. V. Bovdi, A. Konovalov, and E.N. Marcos, Integral group ring of the Suzuki sporadic simple group, Publ. Math. Debrecen 72 (2008), no. 3-4, 487-503. MR 2406705
  • 10. V. Bovdi, A. Konovalov, R. Rossmanith, and Cs. Schneider, LAGUNA--Lie AlGebras and UNits of group Algebras, Version 3.5.0, 2009, (http://www.cs.st-andrews.ac.uk/ ~alexk/laguna.htm).
  • 11. V. Bovdi, A. Konovalov, and S. Siciliano, Integral group ring of the Mathieu simple group $ M\sb{12}$, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 125-136. MR 2313777
  • 12. J.A. Cohn and D. Livingstone, On the structure of group algebras. I, Canad. J. Math. 17 (1965), 583-593. MR 0179266 (31:3514)
  • 13. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985, Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray. MR 827219 (88g:20025)
  • 14. ECLiPSe Constraint Programming System, Ver. 5.10, (http://www.eclipse-clp.org), 2006.
  • 15. GAP--Groups, Algorithms, and Programming, Version 4.4.12, (http://www.gap-system.org), 2008.
  • 16. I. P. Gent, C. Jefferson, and I. Miguel, Minion: A fast scalable constraint solver, ECAI 2006, 17th European Conference on Artificial Intelligence, August 29 - September 1, 2006, Riva del Garda, Italy, Including Prestigious Applications of Intelligent Systems (PAIS 2006), Proceedings, IOS Press, 2006, pp. 98-102.
  • 17. M. Hertweck, On the torsion units of some integral group rings, Algebra Colloq. 13 (2006), no. 2, 329-348. MR 2208368
  • 18. -, The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra 36 (2008), no. 10, 3585-3588. MR 2458394
  • 19. -, Torsion units in integral group rings of certain metabelian groups, Proc. Edinb. Math. Soc. 51 (2008), no. 2, 363-385. MR 2465913 (2009j:16027)
  • 20. -, Partial augmentations and Brauer character values of torsion units in group rings, Comm. Algebra (to appear, 2007), 1-16, (E-print arXiv:math.RA/0612429v2).
  • 21. C. Jansen, K. Lux, R. Parker, and R. Wilson, An Atlas of Brauer Characters, London Mathematical Society Monographs. New Series, vol. 11, Clarendon Press Oxford University Press, New York, 1995, Appendix 2 by T. Breuer and S. Norton, Oxford Science Publications. MR 1367961 (96k:20016)
  • 22. W. Kimmerle, On the prime graph of the unit group of integral group rings of finite groups, Groups, rings and algebras, Contemp. Math., vol. 420, Amer. Math. Soc., Providence, RI, 2006, pp. 215-228. MR 2279241
  • 23. I. S. Luthar and I. B. S. Passi, Zassenhaus conjecture for $ A\sb 5$, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 1, 1-5. MR 1004634 (90g:20007)
  • 24. Mini-Workshop: Arithmetik von Gruppenringen, Oberwolfach Rep. 4 (2007), no. 4, 3209-3239, Abstracts from the mini-workshop held November 25-December 1, 2007, Organized by E. Jespers, Z. Marciniak, G. Nebe and W. Kimmerle, Oberwolfach Reports. Vol. 4, no. 4. MR 2463649
  • 25. H. Zassenhaus, On the torsion units of finite group rings, Studies in mathematics (in honor of A. Almeida Costa) (Portuguese), Instituto de Alta Cultura, Lisbon, 1974, pp. 119-126. MR 0376747 (51:12922)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 16S34, 20C05, 20D08

Retrieve articles in all journals with MSC (2010): 16S34, 20C05, 20D08


Additional Information

V. A. Bovdi
Affiliation: Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary
Email: vbovdi@math.unideb.hu

E. Jespers
Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium
Email: efjesper@vub.ac.be

A. B. Konovalov
Affiliation: School of Computer Science, University of St Andrews, Jack Cole Building, North Haugh, St Andrews, Fife, KY16 9SX, Scotland
Email: alexk@mcs.st-andrews.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2010-02376-2
Keywords: Zassenhaus conjecture, prime graph, torsion unit, partial augmentation, integral group ring
Received by editor(s): April 27, 2007
Received by editor(s) in revised form: September 7, 2009
Published electronically: June 9, 2010
Additional Notes: The research was supported by OTKA No. K68383, Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish-Polish bilateral agreement BIL2005/VUB/2006, Francqui Stichting (Belgium) grant ADSI107 and The Royal Society of Edinburgh International Exchange Programme
Dedicated: Dedicated to the memory of Professor I. S. Luthar
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society