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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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15

The number of irreducible characters

We devote this chapter to the theorem which states that the number of irreducible characters of a finite group is equal to the number of conjugacy classes of the group, and to some consequences of this theorem. Together with the material from Chapter 14, the theorem provides machinery for investigating characters which is used in the remainder of the book.

Throughout, G is as usual a finite group.

Class functions

15.1 Definition

A class function on G is a function ψ: Gimage such that ψ(x) = ψ(y) whenever x and y are conjugate elements of G (that is, ψ is constant on conjugacy classes).

By Proposition 13.5(2) ...

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