The length of the group algebra of the dihedral group of order $2^k$
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 169-181 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order $2^{k+1} $ over an arbitrary field of characteristic $2$ is equal to $2^{k}$.
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O. V. Markova; M. A. Khrystik. The length of the group algebra of the dihedral group of order $2^k$. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 169-181. http://geodesic.mathdoc.fr/item/ZNSL_2020_496_a12/

[1] L. Babai, Á. Seress, “On the diameter of permutation groups”, Eur. J. Combin., 13:4 (1992), 231–243 | DOI | MR | Zbl

[2] E. Couselo, S. González, V. Markov, C. Martínez, A. Nechaev, “Some constructions of linearly optimal group codes”, Linear Algebra Appl., 433:2 (2010), 356–364 | DOI | MR | Zbl

[3] E. Kouselo, S. Gonsales, V. T. Markov, K. Martines, A. A. Nechaev, “Predstavleniya kodov Rida–Solomona i Rida–Mallera idealami”, Algebra i logika, 51:3 (2012), 297–320 | MR

[4] A. E. Guterman, O. V. Markova, “Dlina gruppovykh algebr grupp nebolshogo razmera”, Zap. nauchn. semin. POMI, 472, 2018, 76–87

[5] A. E. Guterman, O. V. Markova, “The length of the group algebra of the group ${\mathbf Q_8}$”, New Trends in Algebra and Combinatorics, Proceedings of the 3rd International Congress in Algebra and Combinatorics, eds. K.P. Shum, E. Zelmanov, P. Kolesnikov, A. Wong, World Sci., Singapore, 2019, 106–134

[6] A. E. Guterman, O. V. Markova, M. A. Khrystik, On the lengths of group algebras of finite Abelian groups in the semi-simple case, Preprint, 2020

[7] A. E. Guterman, M. A. Khrystik, O. V. Markova, On the lengths of group algebras of finite Abelian groups in the modular case, Preprint, 2020

[8] T. Hurley, “Dimension and Fox subgroups”, Resenhas IME-USP, 5:4 (2002), 293–304 | MR | Zbl

[9] S. A. Jennings, “The structure of the group ring of a $p$-group over a modular field”, Trans. Amer. Math. Sot., 50 (1941), 175–185 | MR

[10] M. A. Khrystik, O. V. Markova, On the length of the group algebra of the dihedral group in the semi-simple case, Preprint, 2020

[11] O. V. Markova, “Verkhnyaya otsenka dliny kommutativnykh algebr”, Matem. sb., 200:12 (2009), 41–62 | Zbl

[12] O. V. Markova, “Funktsiya dliny i matrichnye algebry”, Fund. prikl. matem., 17:6 (2012), 65–173

[13] O. V. Markova, “O svyazi dliny algebry i indeksa nilpotentnosti ee radikala Dzhekobsona”, Matem. zametki, 94:5 (2013), 682–688 | Zbl

[14] C. J. Pappacena, “An upper bound for the length of a finite-dimensional algebra”, J. Algebra, 197 (1997), 535–545 | DOI | MR | Zbl

[15] D. S. Passman, The Algebraic Structure of Group Rings, John Wiley Sons, New York–London–Sydney–Toronto, 1977 | MR | Zbl

[16] A. Paz, “An application of the Cayley–Hamilton theorem to matrix polynomials in several variables”, Linear Multilinear Algebra, 15 (1984), 161–170 | DOI | MR | Zbl

[17] J. N. Roksvold, Applying Jennings theory and the $\mathcal{M}$-series to modular isomorphism problems, Thesis for the degree of Master in Mathematics (Master of Science), Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Oslo, 2011

[18] I. N. Tumaikin, “Bazisnye kody Rida–Mallera kak gruppovye kody”, Fund. prikl. matem., 18:4 (2013), 137–154

[19] I. N. Tumaikin, “Idealy gruppovykh kolets, svyazannye s kodami Rida–Mallera”, Fund. prikl. matem., 21:1 (2016), 211–215