Torsion units in integral group rings of Janko simple groups

Bovdi, V.; Jespers, E.; Konovalov, A.

doi:10.1090/S0025-5718-2010-02376-2

Torsion units in integral group rings of Janko simple groups
HTML articles powered by AMS MathViewer

by V. A. Bovdi, E. Jespers and A. B. Konovalov;

Math. Comp. 80 (2011), 593-615

DOI: https://doi.org/10.1090/S0025-5718-2010-02376-2

Published electronically: June 9, 2010

PDF | Request permission

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

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 (Russian). Translated in J. Soviet Math. 57 (1991), no. 2, 2931–2958. MR 1039822
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
Victor Bovdi and Martin Hertweck, Zassenhaus conjecture for central extensions of $S_5$, J. Group Theory 11 (2008), no. 1, 63–74. MR 2381018, DOI 10.1515/JGT.2008.004
Victor Bovdi and Alexander 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, DOI 10.1017/CBO9780511721212.016
V. A. Bovdi and A. B. Konovalov, Integral group ring of the McLaughlin simple group, Algebra Discrete Math. 2 (2007), 43–53. MR 2364062
V. A. Bovdi and A. B. Konovalov, Integral group ring of the Mathieu simple group $M_{23}$, Comm. Algebra 36 (2008), no. 7, 2670–2680. MR 2422512, DOI 10.1080/00927870802068045
—, Torsion units in integral group ring of Higman-Sims simple group, Studia Sci. Math. Hungar. (to appear, 2009).
V. A. Bovdi, A. B. Konovalov, and S. Linton, Torsion units in integral group ring of the Mathieu simple group $\textrm {M}_{22}$, LMS J. Comput. Math. 11 (2008), 28–39. MR 2379938, DOI 10.1112/S1461157000000516
Victor A. Bovdi, Alexander B. Konovalov, and Eduardo do Nascimento Marcos, Integral group ring of the Suzuki sporadic simple group, Publ. Math. Debrecen 72 (2008), no. 3-4, 487–503. MR 2406705, DOI 10.5486/pmd.2008.4115
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).
V. A. Bovdi, A. B. Konovalov, and S. Siciliano, Integral group ring of the Mathieu simple group $M_{12}$, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 125–136. MR 2313777, DOI 10.1007/BF03031434
James A. Cohn and Donald Livingstone, On the structure of group algebras. I, Canadian J. Math. 17 (1965), 583–593. MR 179266, DOI 10.4153/CJM-1965-058-2
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ 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
ECLiPSe Constraint Programming System, Ver. 5.10, (http://www.eclipse-clp.org), 2006.
GAP—Groups, Algorithms, and Programming, Version 4.4.12, (http://www.gap-system.org), 2008.
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.
Martin Hertweck, On the torsion units of some integral group rings, Algebra Colloq. 13 (2006), no. 2, 329–348. MR 2208368, DOI 10.1142/S1005386706000290
Martin Hertweck, The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra 36 (2008), no. 10, 3585–3588. MR 2458394, DOI 10.1080/00927870802157632
Martin Hertweck, Torsion units in integral group rings of certain metabelian groups, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 2, 363–385. MR 2465913, DOI 10.1017/S0013091505000039
—, 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).
Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. MR 1367961
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, DOI 10.1090/conm/420/07977
I. S. Luthar and I. B. S. Passi, Zassenhaus conjecture for $A_5$, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 1, 1–5. MR 1004634, DOI 10.1007/BF02874643
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 Eric Jespers, Zbigniew Marciniak, Gabriele Nebe and Wolfgang Kimmerle; Oberwolfach Reports. Vol. 4, no. 4. MR 2463649, DOI 10.4171/OWR/2007/55
Hans Zassenhaus, On the torsion units of finite group rings, Studies in mathematics (in honor of A. Almeida Costa) (Portuguese), Inst. Alta Cultura, Lisbon, 1974, pp. 119–126. MR 376747