The Unit Group of the Character Ring

Abstract

Let G be a finite group and R(G) be the character ring of G. We determine the structure of the unit group U(R(G)) of R(G). Since R(G) is commutative, the torsion subgroup and the rank of U(R(G)) need to be determined. A theorem of Saksonov states that the torsion subgroup of U(R(G)) consists of the linear characters of G and their additive inverses. In this article, we give an explicit formula for the rank of U(R(G)).

Key Words:

• Character ring
• Unit group

2010 Mathematics Subject Classification:

• 06B05
• 19A22
• 20C99

ACKNOWLEDGMENTS

The author thanks B. Külshammer for some valuable hints. This article is based on a part of the author's Ph.D. dissertation [7].

Notes

Communicated by A. Turull.