Symplectic commutator subgroups

This paper expands on the work of Douglas Costa and Gordon Keller. Costa and Keller used a structure called the commutator subgroup to find the normal subgroups of special linear groups and symplectic groups. It is often difficult to determine which matrices are in the commutator subgroup. They create a homomorphism on the special linear group which provides us with a formula on the entries of a matrix in that group in order to tell us whether or not a matrix is in the commutator subgroup. This information is not only useful in finding normal subgroups, but in studying power residue symbols. This paper creates a homomorphism on the symplectic group Sp(2, M2(A)) for a commutative ring A that provides us with a formula on the entries of an element of that group which tells us whether or not the matrix is in the commutator subgroup. This homomorphism is also extended to one on Ep(2, M2(A), S) (the elementary group generated by elementaries whose off daigonal entries come from S) for S a particular kind of ideal and Jordan ideal.