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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.

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