In this paper, the length of the group algebra of the direct product of a cyclic group and a cyclic p-group over a field of characteristic p is determined. A general lower bound for the length of a commutative group algebra is proved. It is shown that in the case of the direct product of a cyclic group and a cyclic p-group this bound is sharp.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. E. Guterman, M. A. Khrystik, and O. V. Markova, “On the lengths of group algebras of finite Abelian groups in the modular case,”J. Algebra Appl., 21, No. 6, 2250117—2250130 (2022).
A. E. Guterman and O. V. Markova, “The length of group algebras of small-order groups,” Zap. Nauchn. Semin. POMI, 472, 76–87 (2018); English transl., J. Math. Sci., 240, No. 6, 754–761 (2019).
A. E. Guterman and O. V. Markova, “The length of the group algebra of the group Q8,” in: New Trends in Algebra and Combinatorics. Proceedings of the 3rd International Congress in Algebra and Combinatorics (Ed. by K. P. Shum, E. Zelmanov, P. Kolesnikov, A. Wong), World Sci., Singapore (2019), pp. 106–134.
A. E. Guterman, O. V. Markova, and M. A. Khrystik, “On the lengths of group algebras of finite Abelian groups in the semi-simple case,” J. Algebra Appl., 21, No. 7, 2250140–2250153 (2022).
M. A. Khrystik and O. V. Markova, “The length of the group algebra of the dihedral group of order 2k,” Zap. Nauchn. Semin. POMI, 496, 169–181 (2020); English transl., J. Math. Sci., 255, No. 3, 324–331 (2021).
M. A. Khrystik and O. V. Markova, “On the length of the group algebra of the dihedral group in the semi-simple case,” Commun. Algebra, 50, No. 5, 2223–2232 (2022).
O. V. Markova, “An upper bound for the lengths of commutative algebras,” Mat. Sb., 200, No. 12, 41–62 (2009).
O. V. Markova, “The length function and matrix algebras,” Fund. Prikl. Mat., 17, No. 6, 65–173 (2012).
O. V. Markova, “An example of length computation for a group algebra of a noncyclic Abelian group in the modular case,” Fund. Prikl. Mat., 23, No. 2, 217–229 (2020).
C. J. Pappacena, “An upper bound for the length of a finite-dimensional algebra,” J. Algebra, 197, 535–545 (1997).
A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables,” Linear Multilinear Algebra, 15, 161–170 (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 524, 2023, pp. 166–176.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Khrystik, M.A. Length of the Group Algebra of the Direct Product of a Cyclic Group and a Cyclic P-Group in the Modular Case. J Math Sci 281, 334–341 (2024). https://doi.org/10.1007/s10958-024-07105-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-024-07105-0