1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

In this final chapter we apply the techniques of group cohomology to the representation theory of finite groups. Given G a finite group we know that ℙ(G) is semi-simple for any field of characteristic zero. Consequently, from the Wedderburn theorems there is a decomposition 

$$F(G) = \sum {{M_{ni}}({D_i})} $$

(0.1)

where the D i run over central simple division algebras with center Ki a finite cyclotomic extension of F. The question that we answer here is the determination of all the classes {D i } ∊ B(F) which arise in this way, that is to say, which division algebras occur in the simple components of the group ring of a finite group.

The Schur Subgroup of the Brauer Group

The rational group algebra of a normally monomial group

An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] of a finite group G associated with any complex irreducible character of G is obtained. A complete set of primitive central idempotents and the Wedderburn decomposition of the rational group algebra of a finite metabelian group is also computed.

The rational group algebra of a finite group

In this survey we collect and present the classical and some recent methods to compute the primitive (central) idempotents in semisimple group algebras.

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Idempotents in group algebras

A complex irreducible character χ of a finite group G, with an affording representation ρ, is defined to have the property &#x1d4ab; if, for all g ∈ G, either χ(g) = 0 or all the eigen-values of ρ(g) have the same order. An explicit expression for the primitive central idempotent of the rational group algebra ℚ[G] associated with a complex irreducible character having the property &#x1d4ab; is derived. Several consequences are then obtained.

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Inder Bir S. Passi

Papers

Primitive Central Idempotents in Rational Group Algebras