Quadratic Reciprocity in a Finite Group

The law of quadratic reciprocity is a gem from number theory. In this article we show that it has a natural interpretation that can be generalized to an arbitrary finite group. Our treatment relies almost exclusively on concepts and results known at least a hundred years ago. A key role in our story is played by group characters. Recall that a character χ of a finite Abelian group G is a homomorphism from G into C∗, the multiplicative group of nonzero complex numbers. The set of all distinct characters forms a group under point-wise multiplication that is isomorphic to G. Later we will need the notion of a character defined on an arbitrary finite group G, which is the trace of a finite-dimensional representation of G. A character χ of the group (Z/nZ)∗ of reduced residue classes modulo a positive integer n gives rise to a Dirichlet character modulo n, also denoted by χ, which is the function on the integers defined by