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.

In 1956, Jeśmanowicz conjectured that, for any positive integer $n$, the Diophantine equation ${\left((f^2-g^2)n\right)}^x+{\left((2fg)n\right)}^y={\left((f^2+g^2)n\right)}^z$ has only the positive integral solution $(x,y,z)=(2,2,2)$, where $f$ and $g$ are positive integers with $f>g$, gcd $(f,g)=1$, and $f\not\equiv g(\mathrm{mod}2)$. Let $r=6k+2$, $k\in \mathbb{N}$, $k\ge 25$. In this paper, combining $p$-adic form of Baker method with some detailed computation, we prove that if $n$ satisfies $n\equiv 0,6,9(\mathrm{mod}12)$, $f=g+1$ and $g=2^r-1$, then the conjecture is true.