Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Preface
- 1 GROUP REPRESENTATIONS
- 2 ELEMENTARY PROPERTIES OF GROUP CHARACTERS
- 3 INDUCED CHARACTERS
- 4 PERMUTATION GROUPS
- 5 GROUP-THEORETICAL APPLICATIONS
- 6 ARITHMETIC PROPERTIES OF GROUP CHARACTERS
- 7 REAL REPRESENTATIONS
- APPENDIX
- List of character tables
- Solutions
- Bibliography
- Index
6 - ARITHMETIC PROPERTIES OF GROUP CHARACTERS
Published online by Cambridge University Press: 13 October 2009
- Frontmatter
- Contents
- Preface to the second edition
- Preface
- 1 GROUP REPRESENTATIONS
- 2 ELEMENTARY PROPERTIES OF GROUP CHARACTERS
- 3 INDUCED CHARACTERS
- 4 PERMUTATION GROUPS
- 5 GROUP-THEORETICAL APPLICATIONS
- 6 ARITHMETIC PROPERTIES OF GROUP CHARACTERS
- 7 REAL REPRESENTATIONS
- APPENDIX
- List of character tables
- Solutions
- Bibliography
- Index
Summary
Real character values
The question of real-valued characters was briefly considered in Exercises 4 to 6 of Chapter 2.
In this section we are concerned with a particular element g of a finite group G and an arbitrary character ψ, which need not be irreducible. We have seen in (2.87) (p. 51) that
Hence we have
Proposition 6.1. The character value ψ(g) is real if and only if
We recall that all characters are class functions (Proposition 1.1(ii)): thus
if x and y are conjugate in G, which we write as
Hence if g–1 ∼ g, then equation (6.2) holds for every character ψ, and ψ(g) is therefore real. We shall now show that the converse is also true.
Theorem 6.1. The numbers ψ(g) are real for all characters ψ of a finite group if and only if
Proof. It only remains to prove that (6.3) is a necessary condition for the reality of all ψ(g). Suppose that g–1 and g belong to distinct conjugacy classes Cα and Cβ respectively. Put
Then by (6.1)
In particular, for each irreducible character χ(i) we have that
In this case the character relations of the second kind ((2.42), p. 51) imply that
It is therefore evident that not all the values of can be real if (6.3) is false.
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- Information
- Introduction to Group Characters , pp. 160 - 169Publisher: Cambridge University PressPrint publication year: 1987