Unitary units in group algebras

LetK[G] denote the group algebra of the finite groupG over the non-absolute fieldK of characteristic ≠ 2, and let *:K[G] →K[G] be theK-involution determined byg*=g ⁻¹ for allg ∈G. In this paper, we study the group A = A(K[G]) of unitary units ofK[G] and we classify those groupsG for which A contains no nonabelian free group. IfK is algebraically closed, then this problem can be effectively studied via the representation theory ofK[G]. However, for general fields, it is preferable to take an approach which avoids having to consider the division rings involved. Thus, we use a result of Tits to construct fairly concrete free generators in numerous crucial special cases. The first author’s research was supported in part by Capes and Fapesp - Brazil. The second author’s research was supported in part by NSF Grant DMS-9224662.