Non-isomorphic 2-groups with isomorphic modular group algebras
-
Diego García-Lucas
,
Leo Margolis
und
Ángel del Río
Abstract
We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.
Funding statement: This research is partially supported by Grant 19880/GERM/15 funded by Fundación Séneca of Murcia, and Grant PID2020-113206GB-I00 funded by MCIN/AEI/10.13039/501100011033.
Acknowledgements
The second author would like to thank Mima Stanojkovski for many helpful conversations on the Modular Isomorphism Problem.
References
[1] C. Bagiński, The isomorphism question for modular group algebras of metacyclic p-groups, Proc. Amer. Math. Soc. 104 (1988), no. 1, 39–42. 10.1090/S0002-9939-1988-0958039-8Suche in Google Scholar
[2] C. Bagiński, On the isomorphism problem for modular group algebras of elementary abelian-by-cyclic p-groups, Colloq. Math. 82 (1999), no. 1, 125–136. 10.4064/cm-82-1-125-136Suche in Google Scholar
[3] C. Bagiński and J. Kurdics, The modular group algebras of p-groups of maximal class II, Comm. Algebra 47 (2019), no. 2, 761–771. 10.1080/00927872.2018.1498860Suche in Google Scholar
[4] H. U. Besche, B. Eick and E. O’Brien, SmallGrp: The GAP small groups library, version 1.4.1, https://gap-packages.github.io/smallgrp/, 2019. Suche in Google Scholar
[5] F. M. Bleher, W. Kimmerle, K. W. Roggenkamp and M. Wursthorn, Computational aspects of the isomorphism problem, Algorithmic Algebra and Number Theory (Heidelberg 1997), Springer, Berlin (1999), 313–329. 10.1007/978-3-642-59932-3_16Suche in Google Scholar
[6] A. Bovdi, Gruppovye kol’ca, University of Uzhgorod, Uzhgorod 1974. Suche in Google Scholar
[7] A. Bovdi, The group of units of a group algebra of characteristic p, Publ. Math. Debrecen 52 (1998), no. 1–2, 193–244. 10.5486/PMD.1998.2009Suche in Google Scholar
[8] V. Bovdi, A. Konovalov, R. Rossmanith and C. Schneider, LAGUNA: Lie algebras and units of group algebras, version 3.9.3, https://gap-packages.github.io/laguna/, 2019. Suche in Google Scholar
[9] R. Brauer, Representations of finite groups, Lectures on modern mathematics. Vol. I, Wiley, New York (1963), 133–175. Suche in Google Scholar
[10] O. Broche and Á. del Río, The modular isomorphism problem for two generated groups of class two, Indian J. Pure Appl. Math. (2021), 10.1007/s13226-021-00182-w. 10.1007/s13226-021-00182-wSuche in Google Scholar
[11] O. Broche, D. García-Lucas and Á. del Río, A classification of the finite two-generated cyclic-by-abelian groups of prime power order, preprint (2021), https://arxiv.org/abs/2106.06449. Suche in Google Scholar
[12] J. F. Carlson, Periodic modules over modular group algebras, J. Lond. Math. Soc. (2) 15 (1977), no. 3, 431–436. 10.1112/jlms/s2-15.3.431Suche in Google Scholar
[13] D. A. Craven, Representation theory of finite groups: A guidebook, Universitext, Springer, Cham 2019. 10.1007/978-3-030-21792-1Suche in Google Scholar
[14] E. C. Dade, Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345–348. 10.1007/BF01109886Suche in Google Scholar
[15] W. E. Deskins, Finite Abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35–40. 10.1215/S0012-7094-56-02304-3Suche in Google Scholar
[16] B. Eick, Computing automorphism groups and testing isomorphisms for modular group algebras, J. Algebra 320 (2008), no. 11, 3895–3910. 10.1016/j.jalgebra.2008.05.002Suche in Google Scholar
[17] B. Eick and A. Konovalov, The modular isomorphism problem for the groups of order 512, Groups St. Andrews 2009 in Bath. Volume 2, London Math. Soc. Lecture Note Ser. 388, Cambridge University, Cambridge (2011), 375–383. 10.1017/CBO9780511842474.005Suche in Google Scholar
[18] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154 (2001), no. 1, 115–138. 10.2307/3062112Suche in Google Scholar
[19]
M. Hertweck and M. Soriano,
On the modular isomorphism problem: Groups of order
[20] G. Higman, The units of group-rings, Proc. Lond. Math. Soc. (2) 46 (1940), 231–248. 10.1112/plms/s2-46.1.231Suche in Google Scholar
[21] G. Higman, Units in group rings, Ph.D. thesis, University of Oxford, Oxford 1940. Suche in Google Scholar
[22] S. A. Jennings, The structure of the group ring of a p-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. 10.2307/1989916Suche in Google Scholar
[23] W. Kimmerle, Beiträge zur ganzzahligen Darstellungstheorie endlicher Gruppen, Bayreuth. Math. Schr. (1991), no. 36, 139–139. Suche in Google Scholar
[24] R. L. Kruse and D. T. Price, Nilpotent rings, Gordon and Breach Science, New York 1969. Suche in Google Scholar
[25] M. Linckelmann, Finite-dimensional algebras arising as blocks of finite group algebras, Representations of algebras, Contemp. Math. 705, American Mathematical Society, Providence (2018), 155–188. 10.1090/conm/705/14202Suche in Google Scholar
[26] A. Makasikis, Sur l’isomorphie d’algèbres de groupes sur un champ modulaire, Bull. Soc. Math. Belg. 28 (1976), no. 2, 91–109. Suche in Google Scholar
[27] L. Margolis and T. Moede, The modular isomorphism problem for small groups – revisiting Eick’s algorithm, preprint (2020), https://arxiv.org/abs/2010.07030. 10.1016/j.jaca.2022.100001Suche in Google Scholar
[28] L. Margolis and M. Stanojkovski, On the modular isomorphism problem for groups of class 3, J. Group Theory (2021), 10.1515/jgth-2020-0174. 10.1515/jgth-2020-0174Suche in Google Scholar
[29] G. Navarro and B. Sambale, On the blockwise modular isomorphism problem, Manuscripta Math. 157 (2018), no. 1–2, 263–278. 10.1007/s00229-017-0990-zSuche in Google Scholar
[30] I. B. S. Passi and S. K. Sehgal, Isomorphism of modular group algebras, Math. Z. 129 (1972), 65–73. 10.1007/BF01229543Suche in Google Scholar
[31] D. S. Passman, Isomorphic groups and group rings, Pacific J. Math. 15 (1965), 561–583.10.2140/pjm.1965.15.561Suche in Google Scholar
[32]
D. S. Passman,
The group algebras of groups of order
[33] D. S. Passman, The algebraic structure of group rings, Pure Appl. Math., Wiley-Interscience, New York 1977. Suche in Google Scholar
[34] S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420–426. 10.1090/S0002-9947-1950-0034758-3Suche in Google Scholar
[35] K. Roggenkamp, Subgroup rigidity of p-adic group rings (Weiss arguments revisited), J. Lond. Math. Soc. (2) 46 (1992), no. 3, 432–448. 10.1112/jlms/s2-46.3.432Suche in Google Scholar
[36] K. Roggenkamp and L. Scott, Isomorphisms of p-adic group rings, Ann. of Math. (2) 126 (1987), no. 3, 593–647. 10.2307/1971362Suche in Google Scholar
[37] T. Sakurai, The isomorphism problem for group algebras: A criterion, J. Group Theory 23 (2020), no. 3, 435–445. 10.1515/jgth-2019-0071Suche in Google Scholar
[38]
M. A. M. Salim and R. Sandling,
The modular group algebra problem for groups of order
[39] R. Sandling, The isomorphism problem for group rings: a survey, Orders and their applications (Oberwolfach 1984), Lecture Notes in Math. 1142, Springer, Berlin (1985), 256–288. 10.1007/BFb0074806Suche in Google Scholar
[40] R. Sandling, The modular group algebra of a central-elementary-by-abelian p-group, Arch. Math. (Basel) 52 (1989), no. 1, 22–27. 10.1007/BF01197966Suche in Google Scholar
[41] R. Sandling, The modular group algebra problem for metacyclic p-groups, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1347–1350. 10.1090/S0002-9939-96-03518-6Suche in Google Scholar
[42] L. L. Scott, Defect groups and the isomorphism problem, Représentations linéaires des groupes finis, Astérisque 191–182, Société Mathématique de France, Paris (1990), no. 181–182, 257–262. Suche in Google Scholar
[43] S. K. Sehgal, Topics in group rings, Monogr. Textb. Pure Appl. Math. 50, Marcel Dekker, New York 1978. Suche in Google Scholar
[44] A. Weiss, Rigidity of p-adic p-torsion, Ann. of Math. (2) 127 (1988), no. 2, 317–332. 10.2307/2007056Suche in Google Scholar
[45] A. Whitcomb, The group ring problem, ProQuest LLC, Ann Arbor 1968; Ph.D. thesis, The University of Chicago, Chicago 1968. Suche in Google Scholar
[46]
M. Wursthorn,
Isomorphisms of modular group algebras: An algorithm and its application to groups of order
[47] The GAP Group, GAP – Groups, algorithms, and programming, version 4.10.2, http://www.gap-system.org, 2019. 10.1093/oso/9780190867522.003.0002Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field
- Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)
- Prescribing Ricci curvature on homogeneous spaces
- Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
- Ricci flow on manifolds with boundary with arbitrary initial metric
- Unbounded negativity on rational surfaces in positive characteristic
- Extending torsors on the punctured Spec(Ainf)
- Non-isomorphic 2-groups with isomorphic modular group algebras
- Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)
Artikel in diesem Heft
- Frontmatter
- An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field
- Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)
- Prescribing Ricci curvature on homogeneous spaces
- Self-similar solutions to fully nonlinear curvature flows by high powers of curvature
- Ricci flow on manifolds with boundary with arbitrary initial metric
- Unbounded negativity on rational surfaces in positive characteristic
- Extending torsors on the punctured Spec(Ainf)
- Non-isomorphic 2-groups with isomorphic modular group algebras
- Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)
