Computation of the Orders of Classical Groups Over Finite Commutative Rings

In the present note, we compute the orders of several classes of classical groups over finite commutative rings. Simultaneously, using the order of GL_n, we obtain some Anzahl theorems of vector space over local rings. For any finite commutative ring R (with identity 1), R can be written as a direct product of a finite number of local rings R_i, i.e. , where R_i is a local ring, and the classical group G(R) can be written as . So to determine the cardinality |G(R)|, we must determine |G(R_i)| the cardinality of classical group G over a local ring.