Units in $FD_{2p^m}$

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 $\mathcal{U}(FD_{2p^m})$.