Finite groups with very few character values

Abstract

Finite groups with very few character values are characterized. The following is the main result of this article: A finite non-abelian group has precisely four character values if and only if it is the generalized dihedral group of a non-trivial elementary abelian 3-group. The proof involves the analysis of the centralizers of involutions.

Key Words:

• centralizers of involutions
• character values
• finite group
• generalized dihedral group
• irreducible character

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C15 (20D60)

Acknowledgments

The author would like to thank Shigeo Koshitani and Chris Parker for helpful comments. The author would also like to thank the anonymous referee for suggesting the more group-theoretic proof of Lemma 4.3(i).