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

  • Received: 29 September 2021 Revised: 10 December 2021 Accepted: 22 December 2021 Published: 06 January 2022
  • MSC : 20H05

  • 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.

    Citation: Guangren Sun, Zhengjun Zhao. SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup[J]. AIMS Mathematics, 2022, 7(4): 5305-5313. doi: 10.3934/math.2022295

    Related Papers:

  • 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.



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