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

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 $\mathrm{\Lambda }$ of SL ${}_n(\mathbb{Q})$, where a matrix $B$ belongs to $\mathrm{\Lambda }$ if and only if the conjugate subgroup $B{\mathrm{\Gamma }}_q(n)B^{-1}$ of principal congruence subgroup ${\mathrm{\Gamma }}_q(n)$ 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 $\mathrm{\Lambda }$. We show that LCD can be described by the prime decomposition of $q$. Generally $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ in this paper.