Unitary units in modular group algebras

Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism gmapsto g^{-1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by vmapsto v^{-1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g-1)hbar{g} with g,hin G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary.