A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized uni
ts is always a p-group. We prove that it is never powerful except, of course, when G is abelian.
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerles conjecture on prime graphs.
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 setwi
se 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.
