Some general algorithms. I: Arithmetic groups

An algebraic matrix group of degree n, defined over Q, is a subgroup of GLn(C) which is the set of common zeros in GLn(C) of finitely many polynomials, with rational coefficients, in the n2 matrix entries. We shall also call such a group a Q-group, of degree n. We say that the Q-group is given explicitly if these polynomials are explicitly given. If G is such a group and R is a subring of C, put