Órdenes de algunos Grupos Lineales Modulares Orders of some Modular Linear Groups

In this paper we give the definitions and calculate the orders of some subgroups of GL(m,Zn) similar to the classical linear subgroups of the general linear group over any field. Specifically, the special modular linear group SL(m,Zn), the orthogonal modular group O(m,Zn), the special orthogonal modular group SO(m,Zn) and finally the symplectic modular group Sp(2m,Zn). The technique used consists in reducing the problem to the prime case and then employ the prime decomposition of n, for this we apply some basic results about group isomorphisms and from modular arithmetic.