Unitary Subgroups of commutative group algebras of characteristic two

TL;DR

This paper determines the order of the unitary subgroup of group algebras of finite cyclic 2-groups over fields of characteristic two, and shows that these subgroups are pairwise non-isomorphic.

Contribution

It establishes the order of all involution-induced unitary subgroups for cyclic 2-groups and proves their non-isomorphism, advancing understanding of their algebraic structure.

Findings

Calculated the order of $V_{\circledast}(FG)$ for all involutions.

Proved that all such unitary subgroups are pairwise non-isomorphic.

Extended the classification of unitary subgroups in group algebras of 2-groups.

Abstract

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