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

Nao Imoto; Ryoma Kobayashi

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arXiv · 2212.13181

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

Nao Imoto·Ryoma Kobayashi

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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 $\Gamma_d(n)$.

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Nao Imoto, Ryoma Kobayashi. 2022-12-26. The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$. https://arxiv.org/abs/2212.13181

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