Units in $FD_{2p}$

TL;DR

This paper investigates the structure of *-unitary units in the group algebra of a dihedral group over a finite field, revealing detailed algebraic properties and subgroup structures.

Contribution

It provides the first detailed description of the *-unitary units and maximal p-subgroups in the group algebra of dihedral groups over finite fields.

Findings

Structure of *-unitary units in FD_{2p} determined

Maximal p-subgroup of the unit group characterized

Basis of the center of the maximal p-subgroup computed

Abstract

In this paper, we present the structure of the group of *-unitary units in the group algebra $FD_{2p}$, where $F$ is a finite field of characteristic $p > 2$, $D_{2p}$ is the dihedral group of order $2p$, and * is the canonical involution of the group algebra $FD_{2p}$. We also provide the structure of the maximal $p$-subgroup of the unit group $\mathscr{U}(FD_{2p})$ and compute a basis of its center.