A Remark on the Congruence Subgroup Problem

Abstract

In the theory of congruence subgroups, one usually shows that, under suitable assumptions, the normal closure of the mth power of an elementary unipotent matrix coincides with the full con- gruence subgroup mod m. For applications, it is sometimes useful to study the subgroup generated by the mth powers of the elementary unipotent elements. We give an elementary proof for the fact that in $SL_n(Z)$ for $n > 3$, this subgroup is normal in a suitably defined congruence subgroup of $SL_n(Z)$ .