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On the modular isomorphism problem for groups of class \(3\) and obelisks. (English) Zbl 1490.20006

Let \(G\) and \(H\) be finite \(p\)-groups and \(k\) be a finite field of characteristic \(p\). The modular isomorphism problem (MIP), which is still open, can be stated as follows: Does the isomorphism \(kG\cong kH\) of the modular group algebras of \(G\) and \(H\) imply that the groups \(G\) and \(H\) are isomorphic? Alternatively, if \(H\) is a subgroup of the unit group of \(kG\) such that \(kG\cong kH\), then \(H\) is said to be a group base of \(kG\). The MIP can then be reformulated as follows: are all group bases of a modular group algebra of a finite \(p\)-group isomorphic?
Since an explicit formulation of the MIP was given by Brauer in 1963, the problem has been studied by many researchers. Inspired by new points of view and approaches, the authors address the MIP for certain classes of groups from different directions and techniques. First, they employ the so-called small group algebra of \(kG\) to solve the MIP for two new classes of groups. Among the biggest merits of the use of this construction, two results stand out: Sandling’s solution of the MIP for groups with central elementary abelian derived subgroup, and likewise, M. A. M. Salim and R. Sandling’s solution for groups of order \(p^5\) [J. Aust. Math. Soc., Ser. A 61, No. 2, 229–237 (1996; Zbl 0874.20003)]. Even though the authors show, by means of an example, that the strategy of Salim and Sandling [loc. cit.] cannot be directly transferred to groups of order \(p^6\), the results in the paper under review cover, for \(p\) odd, \(5\) of \(43\) isoclinism classes of groups of order \(p^6\), while \(10\) other classes are covered by already known results. The authors themselves claim that the methods employed can be applied to other families of groups having maximal subgroups of relatively small class.
The paper continues with a new approach to derive properties of the lower central series of a finite \(p\)-group from the structure of the associated modular group algebra. Using original quotient-in-quotient embedding techniques, the authors show that, for certain \(2\)-generated groups of nilpotency class \(3\), the isomorphims types of the members of the lower central series of any group base are determined by \(kG\).
The paper concludes with the study of the MIP for a family of groups named \(p\)-obelisks. A \(p\)-obelisk is a finite non-abelian \(p\)-group \(P\) satisfying \(|P:\gamma_2(P)|=p^2\) and \(P^p=\gamma_3(P)\). These groups are hightlighted by recent computer-aided investigations for groups of order \(5^6\), and in some sense are among the groups of order \(5^6\) that are “closest” to being counterexamples to the MIP. The authors prove that if \(kG\cong KH\) and \(G\) is a \(p\)-obelisk, then so is \(H\).

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20D15 Finite nilpotent groups, \(p\)-groups

Citations:

Zbl 0874.20003

Software:

GAP; SmallGrp
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Full Text: DOI arXiv

References:

[1] 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
[2] C. Bagiński and A. Caranti, The modular group algebras of 𝑝-groups of maximal class, Canad. J. Math. 40 (1988), no. 6, 1422-1435. · Zbl 0665.20003
[3] C. Bagiński and A. Konovalov, The modular isomorphism problem for finite 𝑝-groups with a cyclic subgroup of index p^2, Groups St. Andrews 2005. Vol. 1, London Math. Soc. Lecture Note Ser. 339, Cambridge University, Cambridge (2007), 186-193. · Zbl 1117.20003
[4] 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
[5] 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.
[6] N. Blackburn, Generalizations of certain elementary theorems on 𝑝-groups, Proc. Lond. Math. Soc. (3) 11 (1961), 1-22. · Zbl 0102.01903
[7] 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
[8] R. Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York (1963), 133-175. · Zbl 0124.26504
[9] O. Broche and Á. del Río, The modular isomorphism problem for two generated groups of class two, preprint 2020, https://arxiv.org/abs/2003.13281. · Zbl 1495.16035
[10] W. E. Deskins, Finite Abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35-40. · Zbl 0075.23905
[11] J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-𝑝 Groups, 2nd ed., Cambridge Stud. Adv. Math. 61, Cambridge University, Cambridge, 1999. · Zbl 0934.20001
[12] B. Eick, Computing automorphism groups and testing isomorphisms for modular group algebras, J. Algebra 320 (2008), no. 11, 3895-3910. · Zbl 1163.20005
[13] 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
[14] 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
[15] M. Hertweck and M. Soriano, Parametrization of central Frattini extensions and isomorphisms of small group rings, Israel J. Math. 157 (2007), 63-102. · Zbl 1120.20004
[16] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. · Zbl 0217.07201
[17] R. James, The groups of order p^6 (𝑝 an odd prime), Math. Comp. 34 (1980), no. 150, 613-637. · Zbl 0428.20013
[18] R. James, M. F. Newman and E. A. O’Brien, The groups of order 128, J. Algebra 129 (1990), no. 1, 136-158. · Zbl 0694.20011
[19] 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. · JFM 67.0072.03
[20] L. Margolis and T. Moede, The modular isomorphism problem for small groups – revisiting Eick’s algorithm, preprint (2020), https://arxiv.org/abs/2010.07030.
[21] E. A. O’Brien and M. R. Vaughan-Lee, The groups with order p^7 for odd prime 𝑝, J. Algebra 292 (2005), no. 1, 243-258. · Zbl 1108.20016
[22] I. B. S. Passi and S. K. Sehgal, Isomorphism of modular group algebras, Math. Z. 129 (1972), 65-73. · Zbl 0234.20003
[23] D. S. Passman, The Algebraic Structure of Group Rings, Pure Appl. Math., John Wiley & Sons, New York, 1977. · Zbl 0368.16003
[24] T. Sakurai, The isomorphism problem for group algebras: a criterion, J. Group Theory 23 (2020), no. 3, 435-445. · Zbl 1471.20001
[25] M. A. M. Salim, The isomorphism problem for the modular group algebras of groups of order p^5, Ph.D. thesis, Department of Mathematics, University of Manchester, 1993.
[26] M. A. M. Salim and R. Sandling, The unit group of the modular small group algebra, Math. J. Okayama Univ. 37 (1995), 15-25. · Zbl 0938.16025
[27] 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
[28] M. A. M. Salim and R. Sandling, The modular group algebra problem for small 𝑝-groups of maximal class, Canad. J. Math. 48 (1996), no. 5, 1064-1078. · Zbl 0863.20003
[29] 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
[30] 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
[31] M. Stanojkovski, Intense automorphisms of finite groups, preprint (2017), https://arxiv.org/abs/1710.08979; to appear in Mem. Amer. Math. Soc. · Zbl 1489.20003
[32] M. Suzuki, Group Theory. II, Grundlehren Math. Wiss. 248, Springer, New York, 1986. · Zbl 0586.20001
[33] 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
[34] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.10.2, 2019, http://www.gap-system.org.
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