×

Torsion units for some almost simple groups. (English) Zbl 1389.16040

Summary: We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group \(\text{PSL}(2,11)\). Additionally we prove that the prime graph question is true for the automorphism group of the simple group \(\text{PSL}(2,13)\).

MSC:

16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)

Software:

LAGUNA; GAP
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] V. A. Artamonov, A. A. Bovdi: Integral group rings: Groups of units and classical K-theory. J. Sov. Math. 57 (1991), 2931–2958; translation from Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom. 27 (1989), 3–43. · Zbl 0732.16020
[2] A. Bächle, L. Margolis: Rational conjugacy of torsion units in integral group rings of non-solvable groups. ArXiv: 1305.7419 [math.RT] (2013).
[3] V. Bovdi, A. Grishkov, A. Konovalov: Kimmerle conjecture for the Held and O’Nan sporadic simple groups. Sci. Math. Jpn. 69 (2009), 353–362. · Zbl 1182.16030
[4] V. Bovdi, M. Hertweck: Zassenhaus conjecture for central extensions of S 5. J. Group Theory 11 (2008), 63–74. · Zbl 1143.16032
[5] V. Bovdi, C. Höfert, W. Kimmerle: On the first Zassenhaus conjecture for integral group rings. Publ. Math. 65 (2004), 291–303. · Zbl 1076.16028
[6] V. A. Bovdi, E. Jespers, A. B. Konovalov: Torsion units in integral group rings of Janko simple groups. Math. Comput. 80 (2011), 593–615. · Zbl 1209.16026
[7] V. Bovdi, A. Konovalov: Integral group ring of the Mathieu simple group M 24. J. Algebra Appl. 11 (2012), Article ID 1250016, 10 pages. · Zbl 1247.16032
[8] V. A. Bovdi, A. B. Konovalov: Torsion units in integral group ring of Higman-Sims simple group. Stud. Sci. Math. Hung. 47 (2010), 1–11. · Zbl 1221.16026
[9] V. A. Bovdi, A. B. Konovalov: Integral group ring of Rudvalis simple group. Ukr. Mat. Zh. 61 (2009), 3–13; and Ukr. Math. J. 61 (2009), 1–13. · Zbl 1209.16027
[10] V. A. Bovdi, A. B. Konovalov: Integral group ring of the Mathieu simple group M 23. Commun. Algebra 36 (2008), 2670–2680. · Zbl 1148.16027
[11] V. Bovdi, A. Konovalov: Integral group ring of the first Mathieu simple group. Groups St. Andrews 2005. Vol. I. Selected Papers of the Conference, St. Andrews, 2005 (C. M. Campbell et al., eds.). London Math. Soc. Lecture Note Ser. 339, Cambridge University Press, Cambridge, 2007, pp. 237–245. · Zbl 1120.16025
[12] V. A. Bovdi, A. B. Konovalov: Integral group ring of the McLaughlin simple group. Algebra Discrete Math. 2007 (2007), 43–53. · Zbl 1159.16028
[13] V. A. Bovdi, A. B. Konovalov, S. Linton: Torsion units in integral group rings of Conway simple groups. Int. J. Algebra Comput. 21 (2011), 615–634. · Zbl 1234.16025
[14] V. A. Bovdi, A. B. Konovalov, S. Linton: Torsion units in integral group ring of theMathieu simple group M 22. LMS J. Comput. Math. (electronic only) 11 (2008), 28–39. · Zbl 1225.16017
[15] V. A. Bovdi, A. B. Konovalov, E. D. N. Marcos: Integral group ring of the Suzuki sporadic simple group. Publ. Math. 72 (2008), 487–503. · Zbl 1156.16022
[16] A. Bovdi, A. Konovalov, R. Rossmanith, C. Schneider: LAGUNA–Lie AlGebras and UNits of group Algebras. 2013, http://www.cs.st-andrews.ac.uk/\(\sim\)alexk/laguna.
[17] V. A. Bovdi, A. B. Konovalov, S. Siciliano: Integral group ring of the Mathieu simple group M12. Rend. Circ. Mat. Palermo (2) 56 (2007), 125–136. · Zbl 1125.16020
[18] M. Caicedo, L. Margolis, Á. del Río: Zassenhaus conjecture for cyclic-by-abelian groups. J. Lond. Math. Soc., II. Ser. 88 (2013), 65–78. · Zbl 1282.16040
[19] J. A. Cohn, D. Livingstone: On the structure of group algebras. I. Can. J. Math. 17 (1965), 583–593. · Zbl 0132.27404
[20] J. Gildea: Zassenhaus conjecture for integral group ring of simple linear groups. J. Algebra Appl. 12 (2013), 1350016, 10 pages. · Zbl 1280.16035
[21] M. Hertweck: Zassenhaus conjecture for A 6. Proc. Indian Acad. Sci., Math. Sci. 118 (2008), 189–195. · Zbl 1149.16027
[22] M. Hertweck: Partial augmentations and Brauer character values of torsion units in group rings. http://arxiv.org/abs/math/0612429 (2007).
[23] M. Hertweck: On the torsion units of some integral group rings. Algebra Colloq. 13 (2006), 329–348. · Zbl 1097.16009
[24] M. Hertweck: Contributions to the Integral Representation Theory of Groups. Habilitationsschrift, University of Stuttgart (electronic publication), Stuttgart, 2004, http://elib.uni-stuttgart.de/opus/volltexte/2004/1638.
[25] M. Hertweck, C. R. Höfert, W. Kimmerle: Finite groups of units and their composition factors in the integral group rings of the group PSL(2, q). J. Group Theory 12 (2009), 873–882. · Zbl 1191.16036
[26] C. Höfert, W. Kimmerle: On torsion units of integral group rings of groups of small order. Groups, Rings and Group Rings. Proc. of the Conf., Ubatuba, 2004 (A. Giambruno et al., eds.). Lect. Notes Pure Appl. Math. 248, Chapman & Hall/CRC, Boca Raton, 2006, pp. 243–252. · Zbl 1107.16031
[27] E. Jespers, W. Kimmerle, Z. Marciniak, G. Nebe (eds.): Mini-Workshop: Arithmetic of group rings. Oberwolfach Rep. 4, 2007, pp. 3209–3240. (In German.) · Zbl 1177.16002
[28] W. Kimmerle: On the prime graph of the unit group of integral group rings of finite groups. Groups, Rings and Algebras (W. Chin et al., eds.). Contemp. Math. 420, American Mathematical Society (AMS), Providence, 2006, pp. 215–228. · Zbl 1126.20001
[29] I. S. Luthar, I. B. S. Passi: Zassenhaus conjecture for A 5. Proc. Indian Acad. Sci., Math. Sci. 99 (1989), 1–5. · Zbl 0678.16008
[30] I. S. Luthar, P. Trama: Zassenhaus conjecture for S 5. Commun. Algebra 19 (1991), 2353–2362. · Zbl 0729.16021
[31] K. Roggenkamp, L. Scott: Isomorphisms of p-adic group rings. Ann. Math. (2) 126 (1987), 593–647. · Zbl 0633.20003
[32] M. A. Salim: The prime graph conjecture for integral group rings of some alternating groups. Int. J. Group Theory 2 (2013), 175–185. · Zbl 1301.16045
[33] M. A. M. Salim: Kimmerle’s conjecture for integral group rings of some alternating groups. Acta Math. Acad. Paedagog. Nyházi. (N. S.) (electronic only) 27 (2011), 9–22. · Zbl 1240.16047
[34] M. A. M. Salim: Torsion units in the integral group ring of the alternating group of degree 6. Commun. Algebra 35 (2007), 4198–4204. · Zbl 1161.16023
[35] The GAP Group: GAP-Groups, Algorithms and Programming. Version 4.4, 2006, http:/www.gap-system.org.
[36] A. Weiss: Rigidity of p-adic p-torsion. Ann. Math. (2) 127 (1988), 317–332. · Zbl 0647.20007
[37] H. Zassenhaus: On the torsion units of finite group rings. Studies in mathematics, Lisbon, 1974, pp. 119–126.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.