Unitary units in modular group algebras

Abstract

Letp be a prime,K a field of characteristicp, G a locally finitep-group,KG the group algebra, andV the group of the units ofKG with augmentation 1. The anti-automorphismg→g −1 ofG extends linearly toKG; this extension leavesV setwise invariant, and its restriction toV followed byv→v −1 gives an automorphism ofV. The elements ofV fixed by this automorphism are calledunitary; they form a subgroup. Our first theorem describes theK andG for which this subgroup is normal inV.For each elementg inG, let\(\bar g\) denote the sum (inKG) of the distinct powers ofg. The elements 1+(g-1)\(h\bar g\) withh,hεG are thebicyclic units ofKG. Our second theorem describes theK andG for which all bicyclic units are unitary.