Abstract
We show that the modular isomorphism problem has a positive answer for groups of nilpotency class 2 with cyclic center, i.e., that for such p-groups G and H an isomorphism between the group algebras FG and FH implies an isomorphism of the groups G and H for F the field of p elements. For groups of odd order this implication is also proven for F being any field of characteristic p. For groups of even order we need either to make an additional assumption on the groups or on the field.
Funding source: Agencia Estatal de Investigación
Award Identifier / Grant number: PID2020-113206GB-I00
Funding statement: The first author acknowledges support by Grant PID2020-113206GB-I00 funded by MCIN/AEI and Grant Fundación Séneca 22004/PI/22 and the second by the Spanish Ministry of Science and Innovation, through the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S) and through the Ramon y Cajal grant program.
Acknowledgements
We would like to thank Ángel González Prieto for discussions on how to attack Problem 5.6.
References
[1] C. Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I, J. Reine Angew. Math. 183 (1941), 148–167. 10.1515/crll.1941.183.148Search in Google Scholar
[2]
C. Bagiński and A. Konovalov,
The modular isomorphism problem for finite p-groups with a cyclic subgroup of index
[3] R. Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York (1963), 133–175. Search in Google Scholar
[4] O. Broche and A. del Río, The modular isomorphism problem for two generated groups of class two, Indian J. Pure Appl. Math. 52 (2021), no. 3, 721–728. 10.1007/s13226-021-00182-wSearch in Google Scholar
[5] W. E. Deskins, Finite Abelian groups with isomorphic group algebras, Duke Math. J. 23 (1956), 35–40. 10.1215/S0012-7094-56-02304-3Search in Google Scholar
[6] V. Drensky, The isomorphism problem for modular group algebras of groups with large centres, Representation Theory, Group Rings, and Coding Theory, Contemp. Math. 93, American Mathematical Society, Providence (1989), 145–153. 10.1090/conm/093/1003349Search in Google Scholar
[7] R. H. Dye, On the Arf invariant, J. Algebra 53 (1978), no. 1, 36–39. 10.1016/0021-8693(78)90202-8Search in Google Scholar
[8] B. Eick, Computing automorphism groups and testing isomorphisms for modular group algebras, J. Algebra 320 (2008), no. 11, 3895–3910. 10.1016/j.jalgebra.2008.05.002Search in Google Scholar
[9] D. García-Lucas, The modular isomorphism problem and abelian direct factors, Mediterr. J. Math. 21 (2023), Article No. 18. 10.1007/s00009-023-02557-1Search in Google Scholar
[10] D. García-Lucas and A. del Río, A reduction theorem for the isomorphism problem of group algebras over fields, J. Pure Appl. Algebra 228 (2024), no. 4, Paper No. 107511. 10.1016/j.jpaa.2023.107511Search in Google Scholar
[11] D. García-Lucas, L. Margolis and A. del Río, Non-isomorphic 2-groups with isomorphic modular group algebras, J. Reine Angew. Math. 783 (2022), 269–274. 10.1515/crelle-2021-0074Search in Google Scholar
[12]
M. Hertweck and M. Soriano,
On the modular isomorphism problem: Groups of order
[13] N. Jacobson, Lie Algebras, Interscience Tracts Pure Appl. Math. 10, Interscience, New York, 1962. Search in Google Scholar
[14] 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. 10.1090/S0002-9947-1941-0004626-6Search in Google Scholar
[15] Y. K. Leong, Odd order nilpotent groups of class two with cyclic centre, J. Aust. Math. Soc. 17 (1974), 142–153. 10.1017/S1446788700016724Search in Google Scholar
[16] Y. K. Leong, Finite 2-groups of class two with cyclic centre, J. Aust. Math. Soc. Ser. A 27 (1979), no. 2, 125–140. 10.1017/S1446788700012052Search in Google Scholar
[17] L. Margolis, The modular isomorphism problem: A survey, Jahresber. Dtsch. Math.-Ver. 124 (2022), no. 3, 157–196. 10.1365/s13291-022-00249-5Search in Google Scholar
[18] L. Margolis and M. Stanojkovski, On the modular isomorphism problem for groups of class 3 and obelisks, J. Group Theory 25 (2022), no. 1, 163–206. 10.1515/jgth-2020-0174Search in Google Scholar
[19] N. A. Nachev and T. Z. Mollov, An isomorphism of modular group algebras of finite 2-groups for which the order of the quotient-group with respect to the center is equal to four, Pliska Stud. Math. Bulgar. 8 (1986), 3–20. Search in Google Scholar
[20] I. B. S. Passi and S. K. Sehgal, Isomorphism of modular group algebras, Math. Z. 129 (1972), 65–73. 10.1007/BF01229543Search in Google Scholar
[21]
D. S. Passman,
The group algebras of groups of order
[22]
M. A. M. Salim,
The isomorphism problem for the modular group algebras of groups of order
[23] 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. 10.4153/CJM-1996-055-xSearch in Google Scholar
[24] 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/BFb0074806Search in Google Scholar
[25] 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/BF01197966Search in Google Scholar
[26] H. N. Ward, Some results on the group algebra of a p-group over a prime field, Seminar on Finite Groups and Related Topics, Harvard University, Cambridge (1960), 13–19. Search in Google Scholar
[27]
M. Wursthorn,
Isomorphisms of modular group algebras: an algorithm and its application to groups of order
[28] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2, 2022, https://www.gap-system.org. Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A note on the essential numerical range of block diagonal operators
- Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles
- Schauder estimates for Bessel operators
- Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators
- Multiple normalized solutions for fractional elliptic problems
- L-series of weakly holomorphic quasimodular forms and a converse theorem
- Supercongruences arising from a 7 F 6 hypergeometric transformation formula
- Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications
- On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center
- The quotient set of the quadratic distance set over finite fields
- Donoho–Stark and Price uncertainty principles for a class of q-integral transforms with bounded kernels
- Improved spectral cluster bounds for orthonormal systems
- Multilinear Fourier integral operators on modulation spaces
- One-sided Gorenstein rings
- Multilinear fourier integral operators on modulation spaces
Articles in the same Issue
- Frontmatter
- A note on the essential numerical range of block diagonal operators
- Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles
- Schauder estimates for Bessel operators
- Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators
- Multiple normalized solutions for fractional elliptic problems
- L-series of weakly holomorphic quasimodular forms and a converse theorem
- Supercongruences arising from a 7 F 6 hypergeometric transformation formula
- Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications
- On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center
- The quotient set of the quadratic distance set over finite fields
- Donoho–Stark and Price uncertainty principles for a class of q-integral transforms with bounded kernels
- Improved spectral cluster bounds for orthonormal systems
- Multilinear Fourier integral operators on modulation spaces
- One-sided Gorenstein rings
- Multilinear fourier integral operators on modulation spaces