Unitary Subgroups of commutative group algebras of characteristic two

Abstract

Let $FG$ be the group algebra of a finite $2$-group $G$ over a finite field $F$ of characteristic two and $⊛$ an involution which arises from $G$. The $⊛$-unitary subgroup of $FG$, denoted by $V_{⊛}(FG)$, is defined to be the set of all normalized units $u$ satisfying the property $u^{⊛}=u^{-1}$. In this paper we establish the order of $V_{⊛}(FG)$ for all involutions $⊛$ which arise from $G$, where $G$ is a finite cyclic $2$-group and show that all $⊛$-unitary subgroups of $FG$ are not isomorphic.