arXiv subjects
Publications and source records attributed to M. Riaz.
Coset diagrams have been used to study quotients, orbits, subgroups and structure of the finitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the $G$-orbits of $\QQ^*(\sqrt{p^k})$, $\QQ^*(\sqrt{2p^k})$, $\QQ^*(\sqrt{2^2p^k})$, and in general $\QQ^*(\sqrt{2^lp^k})$, for each $l\geq3$ and $k=2h+1\geq3$, for each odd prime $p$.
It is well known that $G=\langle x,y:x^2=y^3=1\rangle$ represents the modular group $PSL(2,Z)$, where $x:z\rightarrow\frac{-1}{z}, y:z\rightarrow\frac{z-1}{z}$ 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 $α=\frac{a+\sqrt{n}}{c}$ in $ Q^*(\sqrt{n})$ by $α(a,b,c)$. For a fixed integer $s>1$, we say that two elements $α(a,b,c)$, $α'(a',b',c')$ of $Q^*(\sqrt{n})$ are $s$-equivalent if and only if $a\equiv a'(mod~s)$, $b\equiv b'(mod~s)$ and $c\equiv c'(mod~s)$. 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 $G$