Unitary units in modular group algebras of 2-groups

Abstract

We study the unitary subgroupV *(F2 G) in the group algebras F2 Gof 2-groups of maximal class over the field F2of two elements. We show that there does not exist a normal complement to Gin V *(F2 G) if and only if Gis the dihedral group of order 2n(n≥ 5) or the semidihedral group of order 2n(n≥5). We also describe V *(F2 G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V *(F2 G).