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

TL;DR

This paper provides a new minimal generating set for the level $d$ principal congruence subgroup of SL(n;Z) and determines its abelianization without relying on previous results by Lee-Szczarba and Tits.

Contribution

It offers a novel approach to find minimal generators and abelianization of $ extrm{SL}(n;bZ)$'s principal congruence subgroups, independent of prior complex results.

Findings

Established a minimal generating set for $ extrm{Gamma}_d(n)$.

Determined the abelianization of $ extrm{Gamma}_d(n)$.

Presented three new theorems about $ extrm{Gamma}_d(n)$.

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 $\Gamma_d(n)$.