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The abelianization of the level $d$ principal congruence subgroup$\Gamma_d(n)$ of $\textrm{SL}(n;\mathbb{Z})$ was determined by Lee-Szczarba. Bythis 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 usingthe results of Tits and Lee-Szczarba. Moreover, we give three theorems about$\Gamma_d(n)$.

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