A note on the action of $SL(m, mathbb{Z}_n)$ on the ring $mathbb{Z}^m_n$

We know that $mathbb{Z}_n$ is a finite field for a prime number $n$. Let $m,n$ be arbitrary natural numbers and let $mathbb{Z}^m_n= mathbb{Z}_n timesmathbb{Z}_ntimes...timesmathbb{Z}_n$ be the Cartesian product of $m$ rings $mathbb{Z}_n$. In this note, we present the action of $SL(m, mathbb{Z}_n)={A in mathbb{Z}^{m,m}_{n} : det A equiv 1 (modsimn)}$, where $SL(m, mathbb{Z}_n)$ for $ngeq 2$ is a group under matrix multiplication modulo $n$, on the ring $mathbb{Z}^m_n$ as a right multiplication of a row vector of $mathbb{Z}^m_n$ by a matrix of $SL(m, mathbb{Z}_n)$ to determine the orbits of the ring $mathbb{Z}^m_n$. This work is an extension of [1]

Equivalent Binary Quadratic Form and the Extended Modular Group

Extended modular group $bar{Pi}=<R,T,U:R^2=T^2=U^3=(RT)^2=(RU)^2=1>$, where $ R:zrightarrow -bar{z}, sim T:zrightarrowfrac{-1}{z},simU:zrightarrowfrac{-1}{z +1} $, has been used to study some properties of the binary quadratic forms whose base points lie in the point set fundamental region $F_{bar{Pi}}$ (See cite{Tekcan1, Flath}). In this paper we look at how base points have been used in the study of equivalent binary quadratic forms, and we prove that two positive definite forms are equivalent if and only if the base point of one form is mapped onto the base point of the other form under the action of the extended modular group and any positive definite integral form can be transformed into the reduced form of the same discriminant under the action of the extended modular group and extend these results for the subset $QQ^*(sqrt{-n})$ of the imaginary quadratic field $QQ(sqrt{-m})$.

G-subsets and G-orbits of Q(sqrt n) under action of the modular group

It is well known that $G=langle x,y:x^2=y^3=1rangle$ represents the modular group $PSL(2,Z)$, where $x:zrightarrowfrac{-1}{z}, y:zrightarrowfrac{z-1}{z}$ are linear fractional transformations. Let $n=k^2m$, where $k$ is any non zero integer and $m$ is square free positive integer. Then the set $$Q^*(sqrt{n}):={frac{a+sqrt{n}}{c}:a,c,b=frac{a^2-n}{c}in Z~textmd{and}~(a,b,c)=1}$$ is a $G$-subset of the real quadratic field $Q(sqrt{m})$ cite{R9}. We denote $alpha=frac{a+sqrt{n}}{c}$ in $ Q^*(sqrt{n})$ by $alpha(a,b,c)$. For a fixed integer $s>1$, we say that two elements $alpha(a,b,c)$, $alpha(a,b,c)$ of $Q^*(sqrt{n})$ are $s$-equivalent if and only if $aequiv a(mod~s)$, $bequiv b(mod~s)$ and $cequiv c(mod~s)$. The class $[a,b,c](mod~s)$ contains all $s$-equivalent elements of $Q^*(sqrt{n})$ and $E^n_s$ denotes the set consisting of all such classes of the form $[a,b,c](mod~s)$. In this paper we investigate proper $G$-subsets and $G$-orbits of the set $Q^*(sqrt{n})$ under the action of Modular Group $G$