UNITS IN $F_2D_{2p}$

TL;DR

This paper explores the structure of units in the group algebra of dihedral groups over the field with two elements, revealing properties of bicyclic units, unit groups, and their subgroups.

Contribution

It determines the structure of the group generated by bicyclic units and the unitary subgroup within the unit group of the algebra, establishing their normality.

Findings

The group generated by bicyclic units is explicitly characterized.

The structure of the unit group and the unitary subgroup is obtained.

Both the bicyclic units group and the unitary subgroup are normal in the unit group.

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 $\mathcal{U}(F_2D_{2p})$.