On the Bloch groups of finite fields and their quotients by the relation corresponding to a tetrahedral symmetry

Abstract

In this note, we show that the natural homomorphisms between the Bloch groups of finite fields and their extensions of odd degree are injective. Further, we give concrete orders of the quotients of the Bloch groups of finite fields by the relation corresponding to a tetrahedral symmetry. In fact, the results are elementary consequences of known results, but these are practically useful and fundamental in studies of the Dijkgraaf{Witten invariant in Bloch groups of finite fields.