MINIMAL CENTRAL ORTHOGONAL IDEMPOTENTS OF SEMISIMPLE GROUP ALGEBRAS OF METACYCLIC GROUP

Let K be an arbitrary field, whose characteristic does not divide the order of the metacyclic group G. In this paper we make the first step towards our examining the structure of the semisimple group algebra KG, namely - finding a complete system of its minimal central orthogonal idempotents. This is important, because every simple component of the Wedderburn decomposition of KG is a minimal two-sided ideal, which is generated by such an idempotent. For this purpose, in Section 2 we first determine the conjugacy classes in G and we calculate their number. With the restriction, that K is a field of decomposition of G, in Section 3 we find the central orthogonal idempotents of KG and we prove that they are minimal. We finish the examination with Theorem 8, in which without the restriction on the field K we find one complete system of minimal central orthogonal idempotents of KG. In Section 4 we give examples in which for a specific group G and field K we compute explicit expressions for the minimal central orthogonal idempotents of KG.