Abstract
In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of $\Gamma _{0}(m)$ in $PSL(2,\mathbb{R})$. We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties.
Highlights
The suborbital graph is a graph arisen from the transitive group action
An interesting contribution of these studies was that the action of normalizer offers solutions for some congruence equations dealing with the sizes of circuits in the suborbital graph [8]
Without any difficulty, it can be seen that the normalizer N or(23p), like N or(23p2), has the elliptic element
Summary
It is known that studying the idea of a group G acting on a set Ω , we can establish some additional structure on Ω. One of these structures is a graph. The suborbital graph is a graph arisen from the transitive group action. When a group G acts on a set Ω , a typical point α is moved by the elements of G to various other points. The set of these images is called the orbit of α under G.
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