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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
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