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On monomial characters and central idempotents of rational group algebras. (English) Zbl 1081.20001
Let \(\mathbb{Q} G\) be the rational group algebra of a finite group \(G\). If \(G\) is a nilpotent group, then E. Jespers, G. Leal and A. Paques [J. Algebra Appl. 2, No. 1, 57-62 (2003; Zbl 1064.20003)] discovered that every primitive central idempotent of \(\mathbb{Q} G\) is determined by a pair \((H,K)\) of subgroups of \(G\), satisfying suitable conditions, and that the primitive central idempotent of \(\mathbb{Q} G\), associated to \((H,K)\), can be easily computed.
In this paper the authors show that this result can be generalized to monomial groups and that the description of the pair of subgroups leading to a primitive central idempotent can be simplified. Recall that a finite group \(G\) is said to be monomial if every irreducible complex character of \(G\) is induced by a linear character of a subgroup of \(G\). Further, the authors apply this method and describe the primitive central idempotents of \(\mathbb{Q} G\), when \(G\) is an Abelian-by-supersolvable or metabelian finite group. Finally, some examples are also given.

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20C15 Ordinary representations and characters
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