Torsion units for some almost simple groups

Joe Gildea

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 561-574
  • ISSN: 0011-4642

Abstract

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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 PSL ( 2 , 11 ) . Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL ( 2 , 13 ) .

How to cite

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Gildea, Joe. "Torsion units for some almost simple groups." Czechoslovak Mathematical Journal 66.2 (2016): 561-574. <http://eudml.org/doc/280090>.

@article{Gildea2016,
abstract = {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 $\rm PSL(2,11)$. Additionally we prove that the Prime graph question is true for the automorphism group of the simple group $\rm PSL(2,13)$.},
author = {Gildea, Joe},
journal = {Czechoslovak Mathematical Journal},
keywords = {Zassenhaus conjecture; torsion unit; partial augmentation; integral group ring; group ring; integral; rational; Zassenhaus conjecture; Kimmerle conjecture; prime graph; torsion unit; partial augmentation; simple group},
language = {eng},
number = {2},
pages = {561-574},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Torsion units for some almost simple groups},
url = {http://eudml.org/doc/280090},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Gildea, Joe
TI - Torsion units for some almost simple groups
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 561
EP - 574
AB - 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 $\rm PSL(2,11)$. Additionally we prove that the Prime graph question is true for the automorphism group of the simple group $\rm PSL(2,13)$.
LA - eng
KW - Zassenhaus conjecture; torsion unit; partial augmentation; integral group ring; group ring; integral; rational; Zassenhaus conjecture; Kimmerle conjecture; prime graph; torsion unit; partial augmentation; simple group
UR - http://eudml.org/doc/280090
ER -

References

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  1. Artamonov, V. A., Bovdi, A. A., 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 3-43 (1989). (1989) 
  2. Bächle, A., Margolis, L., Rational conjugacy of torsion units in integral group rings of non-solvable groups, ArXiv:1305.7419 [math.RT] (2013). (2013) MR3715687
  3. Bovdi, V., Grishkov, A., Konovalov, A., Kimmerle conjecture for the Held and O'Nan sporadic simple groups, Sci. Math. Jpn. 69 (2009), 353-362. (2009) Zbl1182.16030MR2510100
  4. Bovdi, V., Hertweck, M., 10.1515/JGT.2008.004, J. Group Theory 11 (2008), 63-74. (2008) Zbl1143.16032MR2381018DOI10.1515/JGT.2008.004
  5. Bovdi, V., Höfert, C., Kimmerle, W., On the first Zassenhaus conjecture for integral group rings, Publ. Math. 65 (2004), 291-303. (2004) Zbl1076.16028MR2107948
  6. Bovdi, V. A., Jespers, E., Konovalov, A. B., 10.1090/S0025-5718-2010-02376-2, Math. Comput. 80 (2011), 593-615. (2011) Zbl1209.16026MR2728996DOI10.1090/S0025-5718-2010-02376-2
  7. Bovdi, V., Konovalov, A., 10.1142/S0219498811005427, J. Algebra Appl. 11 (2012), Article ID 1250016, 10 pages. (2012) Zbl1247.16032MR2900886DOI10.1142/S0219498811005427
  8. Bovdi, V. A., Konovalov, A. B., Torsion units in integral group ring of Higman-Sims simple group, Stud. Sci. Math. Hung. 47 (2010), 1-11. (2010) Zbl1221.16026MR2654223
  9. Bovdi, V. A., Konovalov, A. B., 10.1007/s11253-009-0199-8, Ukr. Mat. Zh. 61 (2009), 3-13 and Ukr. Math. J. 61 (2009), 1-13. (2009) Zbl1209.16027MR2562187DOI10.1007/s11253-009-0199-8
  10. Bovdi, V. A., Konovalov, A. B., 10.1080/00927870802068045, Commun. Algebra 36 (2008), 2670-2680. (2008) Zbl1148.16027MR2422512DOI10.1080/00927870802068045
  11. Bovdi, V., Konovalov, A., Integral group ring of the first Mathieu simple group, Groups St. Andrews 2005. Vol. I. Selected Papers of the Conference, St. Andrews, 2005 London Math. Soc. Lecture Note Ser. 339 Cambridge University Press, Cambridge (2007), 237-245 C. M. Campbell et al. (2007) Zbl1120.16025MR2328163
  12. Bovdi, V. A., Konovalov, A. B., Integral group ring of the McLaughlin simple group, Algebra Discrete Math. 2007 (2007), 43-53. (2007) Zbl1159.16028MR2364062
  13. Bovdi, V. A., Konovalov, A. B., Linton, S., 10.1142/S0218196711006376, Int. J. Algebra Comput. 21 (2011), 615-634. (2011) Zbl1234.16025MR2812661DOI10.1142/S0218196711006376
  14. Bovdi, V. A., Konovalov, A. B., Linton, S., 10.1112/S1461157000000516, LMS J. Comput. Math. (electronic only) 11 (2008), 28-39. (2008) Zbl1225.16017MR2379938DOI10.1112/S1461157000000516
  15. Bovdi, V. A., Konovalov, A. B., Marcos, E. D. N., Integral group ring of the Suzuki sporadic simple group, Publ. Math. 72 (2008), 487-503. (2008) Zbl1156.16022MR2406705
  16. Bovdi, A., Konovalov, A., Rossmanith, R., Schneider, C., LAGUNA---Lie AlGebras and UNits of group Algebras, (2013), http://www.cs.st-andrews.ac.uk/ {alexk/laguna}. 
  17. Bovdi, V. A., Konovalov, A. B., Siciliano, S., 10.1007/BF03031434, Rend. Circ. Mat. Palermo (2) 56 (2007), 125-136. (2007) Zbl1125.16020MR2313777DOI10.1007/BF03031434
  18. Caicedo, M., Margolis, L., Río, Á. del, 10.1112/jlms/jdt002, J. Lond. Math. Soc., II. Ser. 88 (2013), 65-78. (2013) MR3092258DOI10.1112/jlms/jdt002
  19. Cohn, J. A., Livingstone, D., 10.4153/CJM-1965-058-2, Can. J. Math. 17 (1965), 583-593. (1965) Zbl0132.27404MR0179266DOI10.4153/CJM-1965-058-2
  20. Gildea, J., 10.1142/S0219498813500163, J. Algebra Appl. 12 (2013), 1350016, 10 pages. (2013) Zbl1280.16035MR3063455DOI10.1142/S0219498813500163
  21. Hertweck, M., 10.1007/s12044-008-0011-y, Proc. Indian Acad. Sci., Math. Sci. 118 (2008), 189-195. (2008) Zbl1149.16027MR2423231DOI10.1007/s12044-008-0011-y
  22. Hertweck, M., Partial augmentations and Brauer character values of torsion units in group rings, http://arxiv.org/abs/math/0612429 (2007). (2007) 
  23. Hertweck, M., 10.1142/S1005386706000290, Algebra Colloq. 13 (2006), 329-348. (2006) Zbl1097.16009MR2208368DOI10.1142/S1005386706000290
  24. Hertweck, M., 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. Hertweck, M., Höfert, C. R., Kimmerle, W., 10.1515/JGT.2009.019, J. Group Theory 12 (2009), 873-882. (2009) MR2582054DOI10.1515/JGT.2009.019
  26. Höfert, C., Kimmerle, W., On torsion units of integral group rings of groups of small order, Groups, Rings and Group Rings. Proc. of the Conf., Ubatuba, 2004 Lect. Notes Pure Appl. Math. 248 Chapman & Hall/CRC, Boca Raton (2006), A. Giambruno et al. 243-252. (2006) Zbl1107.16031MR2226199
  27. Jespers, E., Kimmerle, W., Marciniak, Z., (eds.), G. Nebe, Mini-Workshop: Arithmetic of group rings, German Oberwolfach Rep. 4 (2007), 3209-3240. (2007) Zbl1177.16002MR2463649
  28. Kimmerle, W., 10.1090/conm/420/07977, Groups, Rings and Algebras Contemp. Math. 420 American Mathematical Society (AMS), Providence (2006), 215-228 W. Chin et al. (2006) Zbl1126.20001MR2279241DOI10.1090/conm/420/07977
  29. Luthar, I. S., Passi, I. B. S., 10.1007/BF02874643, Proc. Indian Acad. Sci., Math. Sci. 99 (1989), 1-5. (1989) Zbl0678.16008MR1004634DOI10.1007/BF02874643
  30. Luthar, I. S., Trama, P., 10.1080/00927879108824263, Commun. Algebra 19 (1991), 2353-2362. (1991) MR1123128DOI10.1080/00927879108824263
  31. Roggenkamp, K., Scott, L., Isomorphisms of p-adic group rings, Ann. Math. (2) 126 (1987), 593-647. (1987) Zbl0633.20003MR0916720
  32. Salim, M. A., The prime graph conjecture for integral group rings of some alternating groups, Int. J. Group Theory 2 (2013), 175-185. (2013) Zbl1301.16045MR3065873
  33. Salim, M. A. M., Kimmerle's conjecture for integral group rings of some alternating groups, Acta Math. Acad. Paedagog. Nyházi. (N.S.) (electronic only) 27 (2011), 9-22. (2011) Zbl1240.16047MR2813587
  34. Salim, M. A. M., 10.1080/00927870701545069, Commun. Algebra 35 (2007), 4198-4204. (2007) Zbl1161.16023MR2372329DOI10.1080/00927870701545069
  35. The GAP Group: GAP-Groups, Algorithms and Programming, Version 4.4, 2006, http:/www.gap-system.org. 
  36. Weiss, A., Rigidity of p -adic p -torsion, Ann. Math. (2) 127 (1988), 317-332. (1988) Zbl0647.20007MR0932300
  37. Zassenhaus, H., On the torsion units of finite group rings, Studies in mathematics Lisbon (1974), 119-126. (1974) Zbl0313.16014MR0376747

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