Serial group rings of finite groups. General linear and close groups

Andrei Kukharev ; Gena Puninski

Algebra and discrete mathematics, Tome 20 (2015) no. 1, pp. 115-125.

Résumé

For a given $p$, we determine when the $p$-modular group ring of a group from $\mathrm{GL}(n,q)$, $\mathrm{SL}(n,q)$ and $\mathrm{PSL}(n,q)$-series is serial.

Détail

Keywords: serial ring, group ring, general linear group, special linear group, projective special linear group.

Bibliographie

[1] J. L. Alperin, Local Representation Theory, Cambridge University Press, 1981 | MR

[2] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Springer Graduate Texts in Math., 13, 2nd edition, 1992 | DOI | MR | Zbl

[3] M. Auslander, I. Reiten, S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge, 1995 | MR | Zbl

[4] Y. Baba, K. Oshiro, Classical Artinian Rings, World Scient. Publ., 2009 | Zbl

[5] L. Dornhoff, Group representation theory, Part A: Ordinary Representation Theory, Pure and applied mathematics, 7, New York, 1971 | MR | Zbl

[6] D. Eisenbud, P. Griffith, “Serial rings”, J. Algebra, 17 (1971), 389–400 | DOI | MR | Zbl

[7] W. Feit, The Representation Theory of Finite Groups, Mathematical Library, 25, North Holland, 1982 | MR | Zbl

[8] P. Fong, B. Srinivasan, “Blocks with cyclic defect groups in $\mathrm{GL}(n,q)$”, Bull. Amer. Math. Soc., 3 (1980), 1041–1044 | DOI | MR | Zbl

[9] P. Fong, B. Srinivasan, “Brauer trees in $\mathrm{GL}(n,q)$”, Math. Z., 187 (1984), 81–88 | DOI | MR | Zbl

[10] D. G. Higman, “Indecomposable representations at characteristic $p$”, Duke J. Math., 21 (1954), 377–381 | DOI | MR | Zbl

[11] G. Hiss, K. Lux, Brauer Trees of Sporadic Groups, 1989, Oxford | MR | Zbl

[12] A. Kukharev, “Seriality of group rings of unimodular projective groups”, Proc. of the 71th Scientific Conf. of Students and PhD students of Belarusian State University, Part 1 (Minsk, May 18–21, 2014), 11–14

[13] A. Kukharev, G. Puninski, “Serial group rings of finite groups. $p$-solvability”, Algebra Discrete Math., 16 (2013), 201–216 | MR | Zbl

[14] A. Kukharev, G. Puninski, “The seriality of group rings of alternating and symmetric groups”, Vestnik of Belarusian State University, Mathemathics and Informatics series, 2 (2014), 61–64

[15] A. Kukharev, G. Puninski, “Serial group rings of finite groups. Sporadic simple groups and Suzuki groups”, Notes Research Semin. Steklov Institute Sanct-Petersb., 435, 2015, 73–94

[16] A. Kukharev, G. Puninski, Yu. Volkov, “The seriality of the group ring of a finite group depends only of characteristic of the field”, Notes Research Semin. Steklov Institute Sanct-Petersb., 423, 2014, 57–66

[17] K. Lux, H. Pahlings, Representations of Groups: a Computational Approach, Cambrodge Studies in Advanced Mathmeatics, 124, 2010 | MR | Zbl

[18] K. Morita, “On group rings over a modular field which possess radicals expressible as principal ideal”, Sci. Repts. Tokyo Daigaku, 4 (1951), 177–194 | MR | Zbl

[19] G. Puninski, Serial Rings, Kluwer, 2001 | MR | Zbl

[20] B. Srinivasan, “On the indecomposable representations of a certain class of groups”, Proc. Lond. Math. Soc., 10 (1960), 497–513 | DOI | MR | Zbl

[21] M. Stather, “Constructive Sylow theorems for the classical groups”, J. Algebra, 316 (2007), 536–559 | DOI | MR | Zbl

[22] A. J. Weir, “Sylow $p$-subgroups of the classical groups over finite fields with chacteristic prime to $p$”, Proc. Amer. Math. Soc., 6 (1955), 529–533 | MR | Zbl