Startseite Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup
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Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

  • Shigeo Koshitani EMAIL logo und Caroline Lassueur
VerĂśffentlicht/Copyright: 12. Januar 2016
Journal of Group Theory
Aus der Zeitschrift Journal of Group Theory Band 19 Heft 4

Abstract

We provide a description of the torsion subgroup 𝑇𝑇⁢(G) of the finitely generated abelian group T⁢(G) of endo-trivial kG-modules in the case that G has a dihedral Sylow 2-subgroup of order at least 8 and k is an algebraically closed field of characteristic p=2. Specifically, we prove that 𝑇𝑇⁢(G)≅X⁢(G), the group of one-dimensional kG-modules, except possibly when G/O2′⁢(G)≅𝔄6, the alternating group of degree 6; in which case G may have nine-dimensional simple torsion endo-trivial modules. Our results complement the tame-representation type investigation of endo-trivial modules started by Carlson–Mazza–Thévenaz in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore, we provide a general reduction result, valid at any prime p, to recover the structure of 𝑇𝑇⁢(G) from the structure of 𝑇𝑇⁢(G/H), where H is a normal p′-subgroup of G.

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: Grant-in-Aid for Scientific Research (C)23540007

Award Identifier / Grant number: (C)15K04776

Funding source: Schweizerischer Nationalfonds zur FĂśrderung der Wissenschaftlichen Forschung

Award Identifier / Grant number: Fellowship for Prospective Researchers PBELP2_-143516

Funding statement: The first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)23540007, 2011–2014 and (C)15K04776, 2015–2018. The second author acknowledges partial financial support by SNSF Fellowship for Prospective Researchers PBELP143516.

The authors are grateful to Ron Solomon for answering a question on finite groups, to Gunter Malle for useful comments on a preliminary version of this text. They also wish to thank Burkhard Külshammer, Geoffrey Robinson, Andrei Marcus, and Rebecca Waldecker for useful discussions. The first author is grateful to the Swiss National Foundation of Sciences for supporting his visit at the TU Kaiserslautern in September 2014 through the second author’s Fellowship for Prospective Researchers PBELP2_143516, and also to Gunter Malle for supporting his visit at the TU Kaiserslautern in May 2015.

References

[1] Auslander M. and Carlson J. F., Almost-split sequences and group rings, J. Algebra 103 (1986), 122–140. 10.1016/0021-8693(86)90173-0Suche in Google Scholar

[2] Bender H., Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt, J. Algebra 17 (1971), 527–554. 10.1016/0021-8693(71)90008-1Suche in Google Scholar

[3] BonnafĂŠ C., Representations of SL2(Fq), Springer, London, 2011. 10.1007/978-0-85729-157-8Suche in Google Scholar

[4] Bosma W., Cannon J. and Playoust C., The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. 10.1006/jsco.1996.0125Suche in Google Scholar

[5] Breuer T., The GAP Character Table Library, Version 1.2.2, GAP package, http://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib. Suche in Google Scholar

[6] Carlson J. F., Endotrivial modules, Recent Developments in Lie Algebras, Groups and Representation Theory, Proc. Sympos. Pure Math. 86, American Mathematical Society, Providence (2012), 99–111. 10.1090/pspum/086/1412Suche in Google Scholar

[7] Carlson J. F., Mazza N. and Nakano D., Endotrivial modules for finite groups of Lie type, J. Reine Angew. Math. 595 (2006), 93–119. 10.1515/CRELLE.2006.045Suche in Google Scholar

[8] Carlson J. F., Mazza N. and Nakano D., Endotrivial modules for the symmetric and the alternating groups, Proc. Edinb. Math. Soc. (2) 52 (2009), 45–66. 10.1017/S0013091506001179Suche in Google Scholar

[9] Carlson J. F., Mazza N. and Thévenaz J., Endotrivial modules for p-solvable groups, Trans. Amer. Math. Soc. 363 (2011), 4979–4996. 10.1090/S0002-9947-2011-05307-9Suche in Google Scholar

[10] Carlson J. F., Mazza N. and Thévenaz J., Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup, J. Eur. Math. Soc. (JEMS) 15 (2013), 157–177. 10.4171/JEMS/358Suche in Google Scholar

[11] Carlson J. F., Mazza N. and Thévenaz J., Torsion-free endotrivial modules, J. Algebra 398 (2014), 413–433. 10.1016/j.jalgebra.2013.01.020Suche in Google Scholar

[12] Carlson J. F. and Thévenaz J., Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000), 303–335. 10.1023/A:1009988424910Suche in Google Scholar

[13] Carlson J. F. and Thévenaz J., The torsion group of endotrivial modules, Algebra Number Theory 9 (2015), 749–765. 10.2140/ant.2015.9.749Suche in Google Scholar

[14] Conway J  H., Curtis R. T., Norton S  P., Parker R. A. and Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford, 1985. Suche in Google Scholar

[15] Dade E. C., Endo-permutation modules over p-groups. I and II, Ann. of Math. (2) 107 (1978), 459–494, and Ann. of Math. (2) 108 (1978), 317–346. 10.2307/1971169Suche in Google Scholar

[16] Gorenstein D., Finite Group Theory, Harper and Row, New York, 1968. Suche in Google Scholar

[17] Gorenstein D., The Classification of Finite Simple Groups Vol. 1: Groups of Noncharacteristic 2 Type, Plenum Press, New York, 1983. 10.1007/978-1-4613-3685-3Suche in Google Scholar

[18] Gorenstein D. and Walter J. H., The characterization of finite groups with dihedral Sylow 2-subgroups. I, II, III, J. Algebra 2 (1965), 85–151., 218–270, 354–393. 10.1016/0021-8693(65)90027-XSuche in Google Scholar

[19] Jones M. R., Some inequalities for the multiplicator of a finite group II, Proc. Amer. Math. Soc. 45 (1974), 167–172. 10.1090/S0002-9939-1974-0352254-2Suche in Google Scholar

[20] Karpilovski G., Projective Representations of Finite Groups, Dekker, New York, 1985. Suche in Google Scholar

[21] Koshitani S., A remark on blocks with dihedral defect groups in solvable groups, Math. Z. 179 (1982), 401–406. 10.1007/BF01215342Suche in Google Scholar

[22] Koshitani S. and Lassueur C., Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups, Manuscripta Math. 148 (2015), 265–282. 10.1007/s00229-015-0739-5Suche in Google Scholar

[23] Landrock P., Finite Group Algebras and Their Modules, London Math. Soc. Lecture Notes 84, Cambridge University Press, Cambridge, 1983. 10.1017/CBO9781107325524Suche in Google Scholar

[24] Lassueur C. and Malle G., Simple endotrivial modules for the linear, unitary and exceptional groups, Math. Z. 280 (2015), 1047–1074. 10.1007/s00209-015-1465-0Suche in Google Scholar

[25] Lassueur C., Malle G. and Schulte E., Simple endotrivial modules for quasi-simple groups, J. Reine Angew. Math. (2013), 10.15.15/crelle-2013-0100. 10.15.15/crelle-2013-0100Suche in Google Scholar

[26] Lassueur C. and Mazza N., Endotrivial modules for the Schur covers of the symmetric and alternating groups, Algebr. Represent. Theory 18 (2015), 1321–1335. 10.1007/s10468-015-9542-ySuche in Google Scholar

[27] Lassueur C. and Mazza N., Endotrivial modules for the sporadic simple groups and their covers, J. Pure Appl. Algebra 219 (2015), 4203–4228. 10.1016/j.jpaa.2015.02.014Suche in Google Scholar

[28] Marcus A., Representation Theory of Group Graded Algebras, Science Publishers, New York, 1999. Suche in Google Scholar

[29] Mazza N. and Thévenaz J., Endotrivial modules in the cyclic case, Arch. Math. 89 (2007), 497–503. 10.1007/s00013-007-2365-2Suche in Google Scholar

[30] Morita K., On group rings over a modular field which possess radicals expressible as principal ideals, Sci. Rep. Tokyo Bunrika Daigaku A4 (1951), 177–194. Untagged page number? Suche in Google Scholar

[31] Nagao H. and Tsushima Y., Representations of Finite Groups, Academic Press, New York, 1989. Suche in Google Scholar

[32] Navarro G. and Robinson G. R., On endo-trivial modules for p-solvable groups, Math. Z. 270 (2012), 983–987. 10.1007/s00209-010-0835-xSuche in Google Scholar

[33] Robinson G. R., On simple endotrivial modules, Bull. Lond. Math. Soc. 43 (2011), 712–716. 10.1112/blms/bdr006Suche in Google Scholar

[34] Scott L. L., Modular permutation representations, Trans. Amer. Math. Soc. 175 (1973), 101–121. 10.1090/S0002-9947-1973-0310051-1Suche in Google Scholar

[35] Suzuki M., On a class of doubly transitive groups II, Ann. of Math. (2) 79 (1964), 514–589. 10.2307/1970408Suche in Google Scholar

[36] Suzuki M., Group Theory II, Springer, Berlin, 1986. 10.1007/978-3-642-86885-6Suche in Google Scholar

[37] ThĂŠvenaz J., G-Algebras and Modular Representation Theory, Clarendon Press, Oxford, 1995. Suche in Google Scholar

[38] Thévenaz J., Endo-permutation modules, a guided tour, Group Representation Theory, EPFL Press, Lausanne (2007), 115–146. Suche in Google Scholar

[39] Walter J. H., The characterization of finite groups with abelian Sylow 2-subgroups, Ann. of Math. (2) 89 (1969), 405–514. 10.2307/1970648Suche in Google Scholar

[40] R. Wilson , J. Thackray , R. Parker , F. Noeske , J. MĂźller , F. LĂźbeck , C. Jansen , G. Hiss and T. Breuer , The Modular Atlas Project, http://www.math.rwth-aachen.de/~MOC. Suche in Google Scholar

[41] Yamazaki K., On projective representations and ring extensions of finite groups, J. Fac. Sci. Univ. Tokyo 10 (1964), 147–195. Suche in Google Scholar

Received: 2015-10-22
Published Online: 2016-1-12
Published in Print: 2016-7-1

Š 2016 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Codegrees and nilpotence class of p-groups
  3. On finite 2-equilibrated p-groups
  4. Minimal dimension of faithful representations for p-groups
  5. On the Christensen–Wang bounds for the ghost number of a p-group algebra
  6. A family of non-Schurian p-Schur rings over groups of order p3
  7. Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup
  8. Symmetric powers of Nat SL(2,𝕂)
  9. Nested GVZ-groups
  10. On some groups generated by involutions
  11. Thompson’s conjecture for finite simple groups of Lie type Bn and Cn
https://doi.org/10.1515/jgth-2015-0044

Artikel in diesem Heft

  1. Frontmatter
  2. Codegrees and nilpotence class of p-groups
  3. On finite 2-equilibrated p-groups
  4. Minimal dimension of faithful representations for p-groups
  5. On the Christensen–Wang bounds for the ghost number of a p-group algebra
  6. A family of non-Schurian p-Schur rings over groups of order p3
  7. Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup
  8. Symmetric powers of Nat SL(2,𝕂)
  9. Nested GVZ-groups
  10. On some groups generated by involutions
  11. Thompson’s conjecture for finite simple groups of Lie type Bn and Cn
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