M. Ashiq and Q. Mushtaq
Actions of a subgroup of the modular
group on an imaginary quadratic field
The imaginary quadratic fields are defined by the
set $\{a+b\sqrt{-n}:a,b\in Q\}$ and are denoted by $Q(\sqrt{-n}),$
where n is a square-free positive integer. In this paper we have
proved that if $\alpha =\frac{a+\sqrt{-n}}{c}\in Q^{\ast
}(\sqrt{-n})=\{\frac{a+\sqrt{-n}}{c}:a,\frac{a^{2}+n}{c},c\in
Z,c\neq 0\},$ then n does not change its value in the orbit
$\alpha G,$ where $G=
$. Also we show that
the number of orbits of $Q^{\ast}(\sqrt{-n})$ under the action
of G are $2[d(n)+2d(n+1)-6]$ and $ 2[d(n)+2d(n+1)-4]$ according
to n is odd or even, except for n=3 for which there are
exactly eight orbits. Also, the action of G on $Q^{\ast}(\sqrt{-n})$ is always intransitive.
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