Linear Algebra Estimates

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.