On the Surjectivity of Certain Maps IV: For Congruence Ideal Subgroups of Type $A_k$ and $C_k$

Abstract: For a positive integer $k$, we extend the surjectivity results from special linear groups (Type $A_k$) and symplectic linear groups (Type $C_k$) onto product of generalized projective spaces by associating the rows or columns, to certain congruence ideal subgroups of special linear groups and symplectic linear groups.