publication . Article . Preprint . 2016

The status of the Zassenhaus conjecture for small groups

Allen Herman; Leo Margolis; Alexander Konovalov;
Open Access English
  • Published: 31 Aug 2016 Journal: Experimental Mathematics, volume 27, issue 4, pages 431-436 (issn: 1058-6458, eissn: 1944-950X, Copyright policy)
  • Country: United Kingdom
Abstract
We identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP$^+$ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the latti...
Subjects
free text keywords: Mathematics - Rings and Algebras, 16U60, 16S34, 20C05, 20C10, Mathematics - Group Theory, Mathematics - Representation Theory, Integral group ring, Group of units, Zassenhaus conjecture, QA Mathematics, T-NDAS, QA, General Mathematics
Related Organizations
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
,
EC| ZC
Project
ZC
Torsion units of integral group rings
  • Funder: European Commission (EC)
  • Project Code: 705112
  • Funding stream: H2020 | MSCA-IF-EF-ST
21 references, page 1 of 2

[1] A. B¨achle, A. Herman, A. Konovalov, L. Margolis, and G. Singh, ZCTU Source Code, https://github.com/drallenherman/ZCTU-Source-Code

[2] A. B¨achle and L. Margolis, HeLP - A GAP-package for torsion units in integral group rings, arXiv:1507.08174v3 [math.RT].

[3] A. B¨achle and L. Margolis, Rational conjugacy of torsion units in integral group ring of non-solvable groups, Proc. Edinburgh Math. Soc., to appear; arXiv:1305.7419v3[mathRT].

[4] V. Bovdi and M. Hertweck, Zassenhaus conjecture for central extensions of Sn, J. Group Theory, 11 (1), (2008), 6374.

[5] M. Caicedo, L. Margolis, and A´ . del R´ıo, Zassenhaus conjecture for cyclic-by-abelian groups, J. London Math. Soc., (2), 88 (1), (2013), 65-78.

[6] M. Dokuchaev and S. Juriaans, Finite subgroups in integral group rings, Can. J. Math., 48 (6), 1996), 1170-1179.

[7] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.7.7; 2015, http://www.gap-system.org.

[8] A. B¨achle, A. Herman, A. Konovalov, L. Margolis, and G. Singh, ZCTU Source Code, https://github.com/alex-konovalov/ZCTU

[9] A. B¨achle and L. Margolis, HeLP - A GAP-package for torsion units in integral group rings, arXiv:1507.08174v3 [math.RT].

[10] A. Herman and G. Singh, Revisiting the Zassenhaus Conjecture on torsion units for the integral group rings of small groups, Proceedings. Math. Sciences. Indian Academy of Sciences, 125 (2), (2015), 167- 172.

[11] M. Hertweck, On the torsion units of some integral group rings, Algebra Colloq., 13 (2), (2006), 329-348. [OpenAIRE]

[12] M. Hertweck, Partial augmentations and Brauer character values of torsion units in group rings, arXiv:math/0612429v2 [math.RA].

[13] M. Hertweck, Torsion units in integral group rings of certain metabelian groups, Proc. Edinburgh Math. Soc., 51, (2008), 363-385. [OpenAIRE]

[14] M. Hertweck, The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra, 36 (10), (2008), 3585-3588. [OpenAIRE]

[15] C. Ho¨fert and W. Kimmerle, On torsion units of integral group rings of groups of small order. Groups, rings and group rings, 243252, Lect. Notes Pure Appl. Math., 248, Chapman & Hall/CRC, Boca Raton, FL, 2006.

21 references, page 1 of 2
Abstract
We identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP$^+$ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the latti...
Subjects
free text keywords: Mathematics - Rings and Algebras, 16U60, 16S34, 20C05, 20C10, Mathematics - Group Theory, Mathematics - Representation Theory, Integral group ring, Group of units, Zassenhaus conjecture, QA Mathematics, T-NDAS, QA, General Mathematics
Related Organizations
Funded by
NSERC
Project
  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
,
EC| ZC
Project
ZC
Torsion units of integral group rings
  • Funder: European Commission (EC)
  • Project Code: 705112
  • Funding stream: H2020 | MSCA-IF-EF-ST
21 references, page 1 of 2

[1] A. B¨achle, A. Herman, A. Konovalov, L. Margolis, and G. Singh, ZCTU Source Code, https://github.com/drallenherman/ZCTU-Source-Code

[2] A. B¨achle and L. Margolis, HeLP - A GAP-package for torsion units in integral group rings, arXiv:1507.08174v3 [math.RT].

[3] A. B¨achle and L. Margolis, Rational conjugacy of torsion units in integral group ring of non-solvable groups, Proc. Edinburgh Math. Soc., to appear; arXiv:1305.7419v3[mathRT].

[4] V. Bovdi and M. Hertweck, Zassenhaus conjecture for central extensions of Sn, J. Group Theory, 11 (1), (2008), 6374.

[5] M. Caicedo, L. Margolis, and A´ . del R´ıo, Zassenhaus conjecture for cyclic-by-abelian groups, J. London Math. Soc., (2), 88 (1), (2013), 65-78.

[6] M. Dokuchaev and S. Juriaans, Finite subgroups in integral group rings, Can. J. Math., 48 (6), 1996), 1170-1179.

[7] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.7.7; 2015, http://www.gap-system.org.

[8] A. B¨achle, A. Herman, A. Konovalov, L. Margolis, and G. Singh, ZCTU Source Code, https://github.com/alex-konovalov/ZCTU

[9] A. B¨achle and L. Margolis, HeLP - A GAP-package for torsion units in integral group rings, arXiv:1507.08174v3 [math.RT].

[10] A. Herman and G. Singh, Revisiting the Zassenhaus Conjecture on torsion units for the integral group rings of small groups, Proceedings. Math. Sciences. Indian Academy of Sciences, 125 (2), (2015), 167- 172.

[11] M. Hertweck, On the torsion units of some integral group rings, Algebra Colloq., 13 (2), (2006), 329-348. [OpenAIRE]

[12] M. Hertweck, Partial augmentations and Brauer character values of torsion units in group rings, arXiv:math/0612429v2 [math.RA].

[13] M. Hertweck, Torsion units in integral group rings of certain metabelian groups, Proc. Edinburgh Math. Soc., 51, (2008), 363-385. [OpenAIRE]

[14] M. Hertweck, The orders of torsion units in integral group rings of finite solvable groups, Comm. Algebra, 36 (10), (2008), 3585-3588. [OpenAIRE]

[15] C. Ho¨fert and W. Kimmerle, On torsion units of integral group rings of groups of small order. Groups, rings and group rings, 243252, Lect. Notes Pure Appl. Math., 248, Chapman & Hall/CRC, Boca Raton, FL, 2006.

21 references, page 1 of 2
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