zbMATH — the first resource for mathematics

Wedderburn decomposition of finite group algebras. (English) Zbl 1111.20005
This paper is concerned with the computation of the Wedderburn decomposition of a semisimple group algebra $$\mathbb{F} G$$, where $$\mathbb{F}$$ is a finite field and $$G$$ a finite group of order prime to $$|\mathbb{F}|$$. There are numerous applications in coding theory. In terms of so-called ‘strongly Shoda pairs $$(K,H)$$’ of $$G$$, which also appear in [A. Olivieri, Á. del Río, J. J. Simón, Commun. Algebra 32, No. 4, 1531-1550 (2004; Zbl 1081.20001)], primitive central idempotents of $$\mathbb{F} G$$ are explicitly given, together with the structural data of the corresponding simple component (i.e., the matrix degree and the centre field).
Getting all primitive central idempotents (so enough strongly Shoda pairs) requires the additional hypothesis that $$G$$ is Abelian-by-supersolvable. In the case when $$G$$ is even metabelian, these pairs $$(K,H)$$ of subgroups $$K,H\leq G$$ suffice: 1. $$K$$ is maximal in $$\{B\leq G:A\leq B\;\&\;B'\leq H\leq B\}$$ where $$A\supset G'$$ is a maximal Abelian subgroup, and 2. $$K/H$$ is cyclic.

MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings
Full Text:
References:
 [1] Berman, S.D., On the theory of group codes, Cybernetics, 3, 25-31, (1967) · Zbl 0216.55803 [2] Charpin, P., The Reed-Solomon code as ideals in a modular algebra, C.R. acad. sci. Paris, ser. I. math., 294, 597-600, (1982) · Zbl 0491.94018 [3] Drensky, V.; Lakatos, P., Monomial ideals group, algebras and error correcting codes, (), 181-188 [4] Evans Sabin, R.; Lomonaco, S.J., Metacyclic error-correcting codes, Aaecc, 6, 191-210, (1995) · Zbl 0823.94017 [5] Jespers, E.; Leal, G.; Paques, A., Central idempotents in rational group algebras of finite nilpotent groups, J. algebra appl., 2, 1, 57-62, (2003) · Zbl 1064.20003 [6] Kelarev, A.V.; Solé, P., Error correcting codes as ideals in group rings, Contemp. math., 273, 11-18, (2001) · Zbl 0983.94043 [7] MacWilliams, F.J., Codes and ideals in group algebras, (), 317-328 [8] Olivieri, A.; del Río, Á.; Simón, J.J., On monomial characters and central idempotents of rational group algebras, Comm. algebra, 32, 4, 1531-1550, (2004) · Zbl 1081.20001 [9] Passman, D., Infinite crossed products, (1989), Academic Press New York · Zbl 0662.16001 [10] Pless, V.S.; Huffman, W.C., Handbook of coding theory, (1998), Elsevier New York · Zbl 0907.94001 [11] A. Poli, Codes dans les algebras de groups abelienes (codes semisimples, et codes modulaires), “Information Theory” (Proceeding of International CNRS Colloquium, Cachan 1977) Colloquium on International CNRS 276 (1978) 261-271. [12] Reiner, I., Maximal orders, (1975), Academic Press New York · Zbl 0305.16001 [13] Renteria, C.; Tapia Recillas, H., Reed – muller codes: an ideal theory approach, Comm. algebra, 25, 401-443, (1997) · Zbl 0868.94045 [14] Evans Sabin, R., On determining all codes in semi-simple group rings, Lect. notes comput. sci., 273, 279-290, (1993) · Zbl 0804.94017 [15] Yamada, T., The Schur subgroup of the Brauer group, lecture notes in mathematics, vol. 397, (1974), Springer Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.