  In this paper we give a generalization of a linear algebra estimate that
occurs in the paper \cite{RS}, by Michael Rosen and Joseph H. Silverman. In
\cite{RS} authors give a bound for the size of a submodule of $(\mathbb{Z}/n
\mathbb{Z})^2$ in terms of a power of the index of any subgroup of automorphism
group of $(\mathbb{Z}/n \mathbb{Z})^2$ which is acting in an abelian way on
that submodule, meaning that given $A$ and $B$ as any two elements in the
automorphism group $AB-BA$ annihilates all elements of the submodule. We will
give a similar estimate for finite submodules of arbitrary dimension $m$ and
subgroups of general linear group acting on them. Later we will derive the
analog of this result for the case of subgroups of the symplectic group acting
on finite submodules in an abelian fashion.
