Units in $FD_{2p^m}$

TL;DR

This paper determines the structure and order of the unit group and its unitary subgroup in the group algebra of a dihedral group over a finite field of characteristic 2, revealing their algebraic properties.

Contribution

It provides the first detailed analysis of the unit group and unitary subgroup structure in $FD_{2p^m}$ for dihedral groups over characteristic 2 fields.

Findings

Computed the order of $U(FD_{2p^m})$

Described the structure of the unit group and unitary subgroup

Proved the normality of the unitary subgroup

Abstract

In this note, we compute the order and provide the structure of the unit group $\mathcal{U}(FD_{2p^m})$ of the group algebra $FD_{2p^m}$, where $F$ is a finite field of characteristic 2 and $D_{2p^m}$ is the dihedral group of order $2p^m$ such that $p$ is an odd prime. Further, we obtain the structure of the unitary subgroup $\mathcal{U}_*(FD_{2p^m})$ with respect to canonical involution * and prove that it is a normal subgroup of the unit group $\mathcal{U}(FD_{2p^m})$.