@article{Sun2022, 
author = {Guangren Sun and Zhengjun Zhao},
title = {SL           n    (      Z    )-normalizer of a principal congruence subgroup},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {4},
pages = {5305-5313},
keywords = {principal congruence subgroup, least common denominator, maximal subgroup},
url = {https://www.sciopen.com/article/10.3934/math.2022295},
doi = {10.3934/math.2022295},
abstract = {Let SL           n    (      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    Λ of SL           n    (      Q    ), where a matrix    B belongs to    Λ if and only if the conjugate subgroup    B      Γ    q    (  n  )      B          −      1       of principal congruence subgroup        Γ    q    (  n  ) of lever    q is contained in modular group SL           n    (      Z    ). The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether B belongs to    Λ. We show that LCD can be described by the prime decomposition of    q. Generally    Λ is not a group, and not even a subsemigroup of SL           n    (      Q    ). Nevertheless, for the case    n  =  2, we present two families of subgroups that are maximal in    Λ in this paper.}
}