The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$

Abstract

The abelianization of the level $d$ principal congruence subgroup $\Gamma_d(n)$ of $\textrm{SL}(n;\mathbb{Z})$ was determined by Lee-Szczarba. By this result and a result of Tits, we can obtain a minimal generating set for $\Gamma_d(n)$. In this paper, we give a minimal generating set for $\Gamma_d(n)$ and determine the abelianization of $\Gamma_d(n)$, without using the results of Tits and Lee-Szczarba. Moreover, we give three theorems about $\Gamma_d(n)$.