The Quadratic Reciprocity Law by Zolotarev and its generalizations

In the paper of Duke and Hopkins (2005), following the approach of E.I. Zolotarev, an analogue of the quadratic reciprocity law for groups was obtained using the Kronecker symbol. We present a short proof of this statement using the Jacobi symbol. The work is mainly of a methodological nature. In this regard, we also provide a proof of the result established in the paper by Frobenius (1914), related to the combinatorial interpretation of the Jacobi symbol.