no. 1
Group algebras in which the socle of the center is an ideal
Brenner, Sofia1, 2; Külshammer, Burkhard2  
1 Department of Mathematics, TU Darmstadt, Germany
2 Institute for Mathematics, Friedrich Schiller University Jena, Germany
Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 1-19.

Let F be a field of characteristic p>0. We study the structure of the finite groups G for which the socle of the center of FG is an ideal in FG and classify the finite p-groups G with this property. Moreover, we give an explicit description of the finite groups G for which the Reynolds ideal of FG is an ideal in FG.

Received:
Accepted:
Published online:
DOI: 10.5802/art.1
Classification: 20C05, 20C20, 16S34
Brenner, Sofia 1, 2; Külshammer, Burkhard 2

1 Department of Mathematics, TU Darmstadt, Germany
2 Institute for Mathematics, Friedrich Schiller University Jena, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Brenner, Sofia; Külshammer, Burkhard. Group algebras in which the socle of the center is an ideal. Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 1-19. doi : 10.5802/art.1. https://art.centre-mersenne.org/articles/10.5802/art.1/

[1] Brauer, Richard Some applications of the theory of blocks of characters of finite groups. I, J. Algebra, Volume 1 (1964) no. 2, pp. 152-167 | DOI | MR | Zbl

[2] Brenner, Sofia The socle of the center of a group algebra, Dissertation, Friedrich-Schiller-Universität Jena, Deutschland (2022)

[3] Brenner, Sofia; Külshammer, Burkhard Ideals in the center of symmetric algebras, Int. Electron. J. Algebra, Volume 34 (2023), pp. 126-151 | DOI | MR | Zbl

[4] Clarke, Robert J. On the Radical of the Centre of a Group Algebra, J. Lond. Math. Soc., II. Ser., Volume 2 (1969) no. 1, pp. 565-572 | DOI | MR | Zbl

[5] Gorenstein, Daniel Finite Groups, Harper’s Series in Modern Mathematics, Harper & Row, 1968 | Zbl

[6] Huppert, Bertram Endliche Gruppen. I, Grundlehren der Mathematischen Wissenschaften, 134, Springer, 1967 | DOI | Zbl

[7] Koshitani, Shigeo A Note on the Radical of the Centre of a Group Algebra, J. Lond. Math. Soc., Volume 18 (1978) no. 2, pp. 243-246 | DOI | MR | Zbl

[8] Külshammer, Burkhard Group-theoretical descriptions of ring-theoretical invariants of group algebras, Representation Theory of Finite Groups and Finite-Dimensional Algebras (Michler, Gerhard O.; Ringel, Claus M., eds.) (Progress in Mathematics), Volume 95, Birkhäuser (1991), pp. 425-442 | DOI | MR | Zbl

[9] Külshammer, Burkhard Centers and radicals of group algebras and blocks, Arch. Math., Volume 114 (2020), pp. 619-629 | DOI | MR | Zbl

[10] Linckelmann, Markus The Block Theory of Finite Group Algebras. Volume 1, London Mathematical Society Student Texts, 91, Cambridge University Press, 2018 | Zbl

[11] Passman, Donald S. The Algebraic Structure of Group Rings, John Wiley & Sons, 1977 | MR | Zbl

[12] The GAP Group GAP – Groups, Algorithms, and Programming, Version 4.10.0, 2018 (https://www.gap-system.org)

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