Connectedness of a suborbital graph for congruence subgroups

Abstract

In this paper, we give necessary and sufficient conditions for the graph H u,n to be connected and a forest.

Connectedness of suborbital graphs for a special subgroup of the modular group

“…A subgroup of Γ is called a congruence group provided it contains the principal congruence group Γ(n).Congruence groups have been of great interest in many fields of mathematics, number theory, group theory, etc. Jones, Singerman and Wicks [1] used the notion of the imprimitive action [3], [4], [9] for a Γ− invariant equivalence relation induced on Q ∪ {∞} by the congruence subgroup Γ 0 (n) = a b c d ∈ Γ : c ≡ 0 (mod n) to obtain some suborbital graphs and examined their connectedness and forest properties.Some applications of this method can be found in the papers, [1], [2].Particularly in [2], [12], [13] and [14] authors give some results about a connection between the periods of elliptic elements of chosen permutation group with the circuits in suborbital graphs of it. In this article we introduce a different invariant equivalence relation by using the congruence subgroup…”

“…to obtain some suborbital graphs and examined their connectedness and forest properties.Some applications of this method can be found in the papers, [1], [2].Particularly in [2], [12], [13] and [14] authors give some results about a connection between the periods of elliptic elements of chosen permutation group with the circuits in suborbital graphs of it. In this article we introduce a different invariant equivalence relation by using the congruence subgroup…”

A remark on some fuchsian groups

“…In [1][2][3][4][5] authors investigate some important classes of modular group subgroups of finite index. It is known that given a group action, representation gives further means to study the object being acted upon, yielding more information about these groups.…”

Suborbital graphs for the Atkin--Lehner group

“…This fact is important because it means that suborbital graphs might have a potential to clarify signature problems taking into account the order of elliptic elements are one of the invariants of signature. Note that it was seen that this relation is just provided unilaterally in [14]. Elliptic elements do not necessarily correspond to circuits of the same order.…”

“…Then similar studies were done for related finitely generated groups.The reader is referred to [2][3][4][5]8,9,[11][12][13][14][15][16] for some relevant previous work on suborbital graphs. Firstly, in [3], it was proved that the elliptic elements in Γ 0 (n) correspond to circuits in the subgraph F u,n of the same order and vice versa.…”