Hostname: page-component-6b88cc9666-ltbkd Total loading time: 0 Render date: 2026-02-18T13:20:34.658Z Has data issue: false hasContentIssue false
  • English
  • Français

On twisted group ring isomorphism problem for p-groups

Part of: Structure and classification of infinite or finite groups Representation theory of groups Rings and algebras arising under various constructions

Published online by Cambridge University Press:  16 February 2024

Gurleen Kaur ,
Surinder Kaur Open the ORCID record for Surinder Kaur [Opens in a new window]  and
Pooja Singla
Gurleen Kaur
Affiliation:
Indian Institute of Science Education and Research Mohali, Knowledge City, Mohali, 140 306, India
Surinder Kaur
Affiliation:
Department of Mathematics, School of Engineering and Sciences, SRM University AP, Amaravati, Andhra Pradesh, 522502, India
Pooja Singla*
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India
*
Corresponding author: Pooja Singla; Email: psingla@iitk.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.

Keywords

twisted group algebrasprojective representationsSchur multiplier

MSC classification

Primary: 16S35: Twisted and skew group rings, crossed products
Secondary: 20C25: Projective representations and multipliers 20E99: None of the above, but in this section

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 2 , May 2024 , pp. 368 - 381
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Berkovich, Y. G., On the order of the commutator subgroup and the Schur multiplier of a finite $p$ -group, J. Algebra 144(2) (1991), 269–272.CrossRefGoogle Scholar
Coleman, D. B., Finite groups with isomorphic group algebras, Trans. Am. Math. Soc. 105 (1962), 1–8.CrossRefGoogle Scholar
Dade, E. C., Deux groupes finis distincts ayant la même algèbre de groupe sur tout corps, Math. Z. 119 (1971), 345–348.CrossRefGoogle Scholar
Ellis, G., On the Schur multiplier of $p$ -groups, Comm. Algebra 27(9) (1999), 4173–4177.CrossRefGoogle Scholar
The GAP Group, Gap – groups, algorithms, and programming, version 4.12.2 (2022).Google Scholar
Green, J. A., On the number of automorphisms of a finite group, Proc. Roy. Soc. London Ser. A 237 (1956), 574–581.Google Scholar
Hatui, S., Characterization of finite $p$ -groups by their Schur multiplier, Proc. Indian Acad. Sci. Math. Sci. 128(4) (2018), Paper No. 49, 9.CrossRefGoogle Scholar
Hatui, S. and Singla, P., On Schur multiplier and projective representations of Heisenberg groups, J. Pure Appl. Algebra 225(11) (2021), Paper No. 106742, 16.CrossRefGoogle Scholar
Hertweck, M., A counterexample to the isomorphism problem for integral group rings, Ann. Math (2) 154(1) (2001), 115–138.CrossRefGoogle Scholar
Hochschild, G. and Serre, J.-P., Cohomology of group extensions, Trans. Am. Math. Soc. 74 (1953), 110–134.CrossRefGoogle Scholar
Isaacs, I. M., Character theory of finite groups (AMS Chelsea Publishing, Providence, RI, 2006), Corrected reprint of the 1976 original [Academic Press, New York; MR0460423].Google Scholar
James, R., The groups of order $p^{6}$ ( $p$ an odd prime), Math. Comp. 34(150) (1980), 613–637.Google Scholar
Karpilovsky, G., Finite groups with isomorphic group algebras, Illinois J. Math. 25(1) (1981), 107–111.CrossRefGoogle Scholar
Karpilovsky, G., Projective representations of finite groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 94 (Marcel Dekker, Inc, New York, 1985).Google Scholar
Margolis, L., The modular isomorphism problem: a survey, Jahresber. Dtsch. Math.-Ver. 124(3) (2022),157–196.CrossRefGoogle Scholar
Margolis, L. and Del Rio, A., Finite subgroups of group rings: a survey (2018), arXiv preprint arXiv: 1809.00718.Google Scholar
Margolis, L. and Schnabel, O., Twisted group ring isomorphism problem, Q. J. Math. 69(4) (2018), 1195–1219.Google Scholar
Margolis, L. and Schnabel, O., The twisted group ring isomorphism problem over fields, Israel J. Math. 238(1) (2020), 209–242.CrossRefGoogle Scholar
Margolis, L. and Schnabel, O., Twisted group ring isomorphism problem and infinite cohomology groups, J. Pure Appl. Algebra 227(4) (2023), Paper No. 107258, 31.CrossRefGoogle Scholar
Niroomand, P., On the order of Schur multiplier of non-abelian $p$ -groups, J. Algebra 322(12) (2009), 4479–4482.CrossRefGoogle Scholar
Niroomand, P., Characterizing finite $p$ -groups by their Schur multipliers, C. R. Math. Acad. Sci. Paris 350(19-20) (2012), 867–870.CrossRefGoogle Scholar
Niroomand, P., A note on the Schur multiplier of groups of prime power order, Ric. Mat. 61(2) (2012), 341–346.CrossRefGoogle Scholar
Niroomand, P., Characterizing finite $p$ -groups by their Schur multipliers, $t(G)=5$ , Math. Rep. (Bucur.) 17(67) (2) (2015), 249–254.Google Scholar
Niroomand, P., Classifying $p$ -groups by their Schur multipliers, Math. Rep. (Bucur.) 20(70) (3) (2018), 279–284.Google Scholar
Perlis, S. and Walker, G. L., Abelian group algebras of finite order, Trans. Am. Math. Soc. 68 (1950), 420–426.CrossRefGoogle Scholar
Polcino Milies, C. and Sehgal, S. K., An introduction to group rings, Algebra and Applications, vol. 1 (Kluwer Academic Publishers, Dordrecht, 2002).CrossRefGoogle Scholar
Sandling, R., The isomorphism problem for group rings: a survey, in Orders and their Applications: Proceedings of a Conference held in Oberwolfach, West Germany June 3-9 (Oxford University Press, 1985), 256–288.CrossRefGoogle Scholar
Schur, J., Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 132 (1907), 85–137.Google Scholar
Zhou, X. M., On the order of Schur multipliers of finite $p$ -groups, Comm. Algebra 22(1) (1994), 1–8.CrossRefGoogle Scholar