On a Theorem of Ritter–Segal

Abstract

An important result of Ritter (1972) and Segal (1972) states that if P is a p-group, then the natural morphism s from the Burnside ring B(P) of P to the Grothendieck ring R _ℚ(P) of rational representations of P:

is surjective, where R is a subgroup of P.

The purpose of this note is to show that if k is a field of positive characteristic p, then for a finite p-group P, the k-vector space k⊗_ℤ R _ℚ(P) is generated by , for subgroups R of P having index lower than p in their normalizer, and , for subgroups Z ⊃ R of P whenever N _P (R)/R is cyclic or quaternion, and Z/R is its unique subgroup of order p.

Key Words:

• Burnside ring
• Finite p-group
• Ring of rational representations

2000 Mathematics Subject Classification:

• 20C15
• 19A22

Notes

Communicated by E. I. Zelmanov.