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Non-isomorphic 2-groups with isomorphic modular group algebras. (English) Zbl 1514.20019

Let \(k\) be a field of characteristic \(p\) and let \(G\) and \(H\) be finite \(p\)-groups. The modular isomorphism problem for group algebras, which was posed by R. Brauer in [Lect. Modern Math. 1, 133–175 (1963; Zbl 0124.26504)], asks whether an isomorphism between the group algebras \(kG\) and \(kH\) implies a group isomorphism between \(G\) and \(H\).
The main result of this impressive paper is the following.
Theorem. There are non-isomorphic finite 2-groups \(G\) and \(H\) such that the group rings of \(G\) and \(H\) over any field of characteristic 2 are isomorphic. In particular, the modular isomorphism problem has a negative answer.
The counterexamples constructed by the authors are so interesting that the reviewer cannot avoid reporting them. Let \(n\) and \(m\) be integers satisfying \(n > m > 2\), then the groups given by the presentations: \[ G=\big \langle x,y,z \mid [x,y]=z,\; x^{2^{n}}=y^{2^{m}}=z^{4}=1, \; z^{x}=z^{-1}, \; z^{y}=z^{-1} \big \rangle\] \[ H=\big \langle a,b,c \mid c=[b,a],\; a^{2^{n}}=b^{2^{m}}=c^{4}=1, \; c^{a}=z^{-1}, \; c^{b}=c \big \rangle\] provide a negative answer to the modular isomorphism problem.
The smallest admissible values, \(n=4\) and \(m=3\), correspond to the groups identified in the library of small groups of GAP as \([512,456]\) and \([512,453]\). These groups have nilpotency class 3 and \(G'\simeq H'\) are cyclic of order 4.
A consequence of the main theorem is the following Corollary: There are isomorphic 2-blocks of finite groups with non-isomorphic defect groups. In particular, the defect group of a 2-block is not determined by its Morita equivalence class over a finite field of characteristic 2.
The modular isomorphism problem remains open for various interesting classes of \(p\)-groups, including groups of nilpotency class 2 and groups of odd order.

MathOverflow Questions:

Breakthroughs in mathematics in 2021

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters

Citations:

Zbl 0124.26504

Software:

LAGUNA; SmallGrp; GAP
PDFBibTeX XMLCite
Full Text: DOI arXiv

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. · Zbl 0663.20006
[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. · Zbl 0943.20007
[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. · Zbl 1472.20003
[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.
[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. · Zbl 0928.20004
[6] A. Bovdi, Gruppovye kol’ca, University of Uzhgorod, Uzhgorod 1974. · Zbl 0339.16004
[7] A. Bovdi, The group of units of a group algebra of characteristic p, Publ. Math. Debrecen 52 (1998), no. 1-2, 193-244. · Zbl 0906.16016
[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.
[9] R. Brauer, Representations of finite groups, Lectures on modern mathematics. Vol. I, Wiley, New York (1963), 133-175. · Zbl 0124.26504
[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. · Zbl 1495.16035 · doi:10.1007/s13226-021-00182-w
[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.
[12] J. F. Carlson, Periodic modules over modular group algebras, J. Lond. Math. Soc. (2) 15 (1977), no. 3, 431-436. · Zbl 0365.20015
[13] D. A. Craven, Representation theory of finite groups: A guidebook, Universitext, Springer, Cham 2019. · Zbl 1446.20002
[14] E. C. Dade, Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345-348. · Zbl 0201.03303
[15] W. E. Deskins, Finite Abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35-40. · Zbl 0075.23905
[16] B. Eick, Computing automorphism groups and testing isomorphisms for modular group algebras, J. Algebra 320 (2008), no. 11, 3895-3910. · Zbl 1163.20005
[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. · Zbl 1231.20002
[18] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154 (2001), no. 1, 115-138. · Zbl 0990.20002
[19] M. Hertweck and M. Soriano, On the modular isomorphism problem: Groups of order 2^6, Groups, rings and algebras, Contemp. Math. 420, American Mathematical Society, Providence (2006), 177-213. · Zbl 1120.20005
[20] G. Higman, The units of group-rings, Proc. Lond. Math. Soc. (2) 46 (1940), 231-248. · JFM 66.0104.04
[21] G. Higman, Units in group rings, Ph.D. thesis, University of Oxford, Oxford 1940. · JFM 66.0104.04
[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. · Zbl 0025.24401
[23] W. Kimmerle, Beiträge zur ganzzahligen Darstellungstheorie endlicher Gruppen, Bayreuth. Math. Schr. (1991), no. 36, 139-139. · Zbl 0728.20005
[24] R. L. Kruse and D. T. Price, Nilpotent rings, Gordon and Breach Science, New York 1969. · Zbl 0198.36102
[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. · Zbl 1436.20014
[26] A. Makasikis, Sur l’isomorphie d’algèbres de groupes sur un champ modulaire, Bull. Soc. Math. Belg. 28 (1976), no. 2, 91-109. · Zbl 0423.20004
[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.
[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. · Zbl 1490.20006 · doi:10.1515/jgth-2020-0174
[29] G. Navarro and B. Sambale, On the blockwise modular isomorphism problem, Manuscripta Math. 157 (2018), no. 1-2, 263-278. · Zbl 1499.20002
[30] I. B. S. Passi and S. K. Sehgal, Isomorphism of modular group algebras, Math. Z. 129 (1972), 65-73. · Zbl 0234.20003
[31] D. S. Passman, Isomorphic groups and group rings, Pacific J. Math. 15 (1965), 561-583. · Zbl 0171.28603
[32] D. S. Passman, The group algebras of groups of order p^4 over a modular field, Michigan Math. J. 12 (1965), 405-415. · Zbl 0134.26304
[33] D. S. Passman, The algebraic structure of group rings, Pure Appl. Math., Wiley-Interscience, New York 1977. · Zbl 0368.16003
[34] S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420-426. · Zbl 0038.17301
[35] K. Roggenkamp, Subgroup rigidity of p-adic group rings (Weiss arguments revisited), J. Lond. Math. Soc. (2) 46 (1992), no. 3, 432-448. · Zbl 0725.20004
[36] K. Roggenkamp and L. Scott, Isomorphisms of p-adic group rings, Ann. of Math. (2) 126 (1987), no. 3, 593-647. · Zbl 0633.20003
[37] T. Sakurai, The isomorphism problem for group algebras: A criterion, J. Group Theory 23 (2020), no. 3, 435-445. · Zbl 1471.20001
[38] M. A. M. Salim and R. Sandling, The modular group algebra problem for groups of order p^5, J. Aust. Math. Soc. Ser. A 61 (1996), no. 2, 229-237. · Zbl 0874.20003
[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. · Zbl 0565.20005
[40] R. Sandling, The modular group algebra of a central-elementary-by-abelian p-group, Arch. Math. (Basel) 52 (1989), no. 1, 22-27. · Zbl 0632.16011
[41] R. Sandling, The modular group algebra problem for metacyclic p-groups, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1347-1350. · Zbl 0844.20003
[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. · Zbl 0727.20002
[43] S. K. Sehgal, Topics in group rings, Monogr. Textb. Pure Appl. Math. 50, Marcel Dekker, New York 1978. · Zbl 0411.16004
[44] A. Weiss, Rigidity of p-adic p-torsion, Ann. of Math. (2) 127 (1988), no. 2, 317-332. · Zbl 0647.20007
[45] A. Whitcomb, The group ring problem, ProQuest LLC, Ann Arbor 1968; Ph.D. thesis, The University of Chicago, Chicago 1968.
[46] M. Wursthorn, Isomorphisms of modular group algebras: An algorithm and its application to groups of order 2^6, J. Symbolic Comput. 15 (1993), no. 2, 211-227. · Zbl 0782.20001
[47] The GAP Group, GAP - Groups, algorithms, and programming, version 4.10.2, http://www.gap-system.org, 2019.
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