UNITS IN $F_2D_{2p}$

Abstract

Let $p$ be an odd prime, $D_{2p}$ be the dihedral group of order 2p, and $F_{2}$ be the finite field with two elements. If * denotes the canonical involution of the group algebra $F_2D_{2p}$, then bicyclic units are unitary units. In this note, we investigate the structure of the group $\mathcal{B}(F_2D_{2p})$, generated by the bicyclic units of the group algebra $F_2D_{2p}$. Further, we obtain the structure of the unit group $\mathcal{U}(F_2D_{2p})$ and the unitary subgroup $\mathcal{U}_*(F_2D_{2p})$, and we prove that both $\mathcal{B}(F_2D_{2p})$ and $\mathcal{U}_*(F_2D_{2p})$ are normal subgroups of $\mathcal{U}(F_2D_{2p})$.