Dirichlet character group (reviewed)

The Dirichlet characters $\chi\colon\Z\to\C$ of modulus $q$ form a group under pointwise multiplication.

In the Conrey labeling system we simply have $\chi_q(n_1,\cdot)\chi_q(n_2,\cdot) = \chi_q(n_1n_2,\cdot)$.

This group is (non-canonically) isomorphic to the multiplicative group $(\Z/q\Z)^\times$.