Wedderburn Decomposition and Structure of a Semi-Simple Dihedral Group Algebra

Abstract

Let \(K\) be an arbitrary field, whose characteristic does not divide the order of the dihedral group \(D_{2 m}\) of order \(2 m\), where \(m\) is odd, and \(K D_{2 m}\) be the group algebra of \(D_{2 m}\) over the field \(K\). The structure of the semisimple dihedral group algebra \(K D_{2 m}\) is examined in this work. We find a complete system of minimal central orthogonal idempotents of the group algebra for this purpose. We define the simple components of \(K D_{2 m}\) and its Wedderburn decomposition through it. The results are as general as possible, i.e. they do not require a finite field.