SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

Abstract

Let SL$ _n(\mathbb{Q}) $ be the set of matrices of order $ n $ over the rational numbers with determinant equal to 1. We study in this paper a subset $ \Lambda $ of SL$ _n(\mathbb{Q}) $, where a matrix $ B $ belongs to $ \Lambda $ if and only if the conjugate subgroup $ B\Gamma_q(n)B^{-1} $ of principal congruence subgroup $ \Gamma_q(n) $ of lever $ q $ is contained in modular group SL$ _n(\mathbb{Z}) $. The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether B belongs to $ \Lambda $. We show that LCD can be described by the prime decomposition of $ q $. Generally $ \Lambda $ is not a group, and not even a subsemigroup of SL$ _n(\mathbb{Q}) $. Nevertheless, for the case $ n = 2 $, we present two families of subgroups that are maximal in $ \Lambda $ in this paper.

Keywords

• principal congruence subgroup
• least common denominator
• maximal subgroup