Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T12:17:40.647Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Structure of Integral Group Rings of Sporadic Groups

Published online by Cambridge University Press:  01 February 2010

Frauke M. Bleher  and
Wolfgang Kimmerle
Frauke M. Bleher
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, U.S.A., [email protected]
Wolfgang Kimmerle
Affiliation:
Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D–70550 Stuttgart, Germany, [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 274 - 306
Copyright
Copyright © London Mathematical Society 2000

References

1. Blanchard, P. F., ‘Exceptional group ring automorphisms for some metabelian groups, II’, Comm. Alg. 25 (1997) 2735–2742.CrossRefGoogle Scholar
2. Bleher, F. M., ‘Zassenhaus-Vermutung und einfache Gruppen’, Diplomarbeit, Universität Stuttgart, 1993.Google Scholar
3. Bleher, F. M., ‘Tensor products and a conjecture of Zassenhaus’, Arch. Math. 64 (1995)289–298.CrossRefGoogle Scholar
4. Bleher, F. M., ‘Automorphismen von Gruppenringen und Blocktheorie’, Dissertation, Universitat Stuttgart, 1995.Google Scholar
5. Bleher, F. M., ‘Finite groups of Lie type of small rank’, Pacific J. Math. 187 (1999) 215–239.CrossRefGoogle Scholar
6. Bleher, F. M., ‘Integral group rings of finite groups of Lie type’, Bull. London Math. Soc. 31 (1999)43–44.CrossRefGoogle Scholar
7. Bleher, F. M., Hiss, G. and Kimmerle, W., ‘Autoequivalences of blocks and a conjecture of Zassenhaus’, J. Pure Appl. Algebra 103 (1995) 23–43.CrossRefGoogle Scholar
8. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, London/New York, 1985).Google Scholar
9. Dokuchaev, M. A. and Juriaans, S. O., ‘Finite subgroups in integral group rings’, Canad. J. Math. 48 (1996) 1170–1179.CrossRefGoogle Scholar
10. Gross, F., ‘Automorphisms which centralize a Sylow p-subgroup’, J. Algebra 11 (1982) 202–233.CrossRefGoogle Scholar
11. Hertweck, M., ‘Eine Lösung des Isomorphieproblems für ganzzahlige Gruppenringe endlicher Gruppen‘, Dissertation, Universität Stuttgart, 1998.Google Scholar
12. Hertweck, M. and Kimmerle, W., ‘On the F*-theorem‘, Proceedings Groups St Andrews in Bath 1997 I (ed. Campbell, C. M., Robertson, E. F., Ruskuc, N. and Smith, G. C.), London Math. Soc. Lect. Note Ser. 260 (Cambridge University Press, 1999)346–352.Google Scholar
13. Huppert, B., Endliche Gruppen I, Grundlehren der Math. Wissenschaften 134 (Springer-Verlag, Berlin-Heidelberg-New York, 1967).CrossRefGoogle Scholar
14. Jansen, C., Lux, K., Parker, R. A. and Wilson, R. A., An atlas of Brauer characters, London Math. Soc. Monographs, New Series 11 (Oxford University Press, Oxford, 1995).Google Scholar
15. Kimmerle, W., ‘Beiträge zur ganzzahligen Darstellungstheorie endlicher Gruppen’, Bayreuther Math. Schriften 36 (1991) 1–139.Google Scholar
16. Kimmerle, W., ‘Class Sums of p-elements’, in: [26] 117–124.CrossRefGoogle Scholar
17. Kimmerle, W., ‘On Automorphisms of ℤG and the Zassenhaus Conjectures’, CMS Conference Proceedings Vol. 18 (1996) 383–397.Google Scholar
18. Kimmerle, W., ‘On the characterization of finite groups by characters’, The atlas of finite groups: ten years on (ed. Curtis, R. and Wilson, R.), London Math. Soc. Lect. Note Ser. 249 (1998) 119–138.CrossRefGoogle Scholar
19. Kimmerle, W., Lyons, R., Sandling, R. and Teague, D., ‘Composition factors from the group ring and Artin's theorem on orders of simple groups’, Proc. London Math. Soc. (3) 60 (1990) 89–122.CrossRefGoogle Scholar
20. Kimmerle, W. and Roggenkamp, K. W., ‘A Sylowlike theorem for integral group rings of finite solvable groups’, Arch. Math. 60 (1993) 1–6.CrossRefGoogle Scholar
21. Klingler, L., ‘Construction of a counterexample to a conjecture of Zassenhaus’, Comm. Algebra 19 (1991) 2303–2330.CrossRefGoogle Scholar
22. Lux, K., ‘Algorithmic methods in modular representation theory’, Habilitationsschrift, RWTH Aachen, 1997.Google Scholar
23. Peterson, G. L. ‘Automorphisms of the integral group ring of Sn’, Proceedings AMS 59 (1976) 14–18.Google Scholar
24. Roggenkamp, K. W., ‘The isomorphism problem for integral group rings of finite groups’, Proceedings of ICM Kyoto 1990 (Springer-Verlag, 1991) 369–380.Google Scholar
25. Roggenkamp, K. W., ‘Units and the isomorphism problem’, Part I of [26].Google Scholar
26. Roggenkamp, K. W. and Taylor, M., Group rings and class groups, DMV-Seminar 18 (Birkhauser, Basel-Boston-Berlin, 1992).Google Scholar
27. Schönert, M. et al. , GAP — groups, algorithms, and programming, first edn (Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1992).Google Scholar
28. Schur, I., ‘Untersuchungen über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen’, J.Math. 132 (1907) 85–137.Google Scholar
29. Scott, L. L., ‘Recent progress on the isomorphism problem’, Proc. Symposia in Pure Math. Vol. 47 (1987) 259–274.CrossRefGoogle Scholar
30. Scott, L. L., ‘Defect groups and the isomorphism problem’, Représentations linéaires des groupes finis, Proc. Colloq. Luminy, France 1988, Astèrisque 181–182 (1990) 257–262.Google Scholar
31. Suzuki, M., Group theory I, Grundlehren der Math. Wissenschaften 247 ( Springer-Verlag, Berlin-Heidelberg-New York, 1982).Google Scholar
32. Waki, K., ‘The projective indecomposable modules for the Higman-Sims group in characteristic 3’, Comm. Algebra 21 (1993) 3475–3487.CrossRefGoogle Scholar