On Suborbital Graphs for the Modular Group

Abstract

This paper proves a conjecture of G. A. Jones, D. Singerman and K. Wicks, that a suborbital graph for the modular group is a forest if and only if it contains no triangles.

Suborbital graphs of a power subgroup of the modular group

“…Then Akbas found certain relationship between the lenghts of circuits in these graphs and periods of elliptic elements of the group Γ 0 (n) [1]. This is important taking into account that the elliptic elements are one of the invariants of the group.…”

Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

“…Hence by Cauchy Frobenius Lemma, the number of orbits of ( ) By applying Cauchy Frobenius Lemma the number of orbits of ( 1,4,7 ) G acting on X Y Z × × are;…”

“…( 1,1,( ab ) ) 1 9 ( 1,( ab ),1 ) 1 9 ( 1,( ab ),( ab ) ) 1 3 ( ( ab ),1,1 ) 1 9 ( ( ab ),( ab ),1 ) 1 3 1 3 ( ( ab ),( ab ),( ab ) ) 1 1 Now applying Theorem 2.2, the number of orbits of ( 1,4,7 ) G acting on =Orb ( 2,5,8 )={ ( 2,5,8 ), ( 2,5,9 ), ( 2,6,8 ), ( 3,5,8 ), ( 2,6,9 ), ( 3,5,9 ), ( 3,6,8 ), ( 3,6,9 )} ∆ From the above, the rank of …”

Suborbital graphs for the Atkin--Lehner group

“…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. 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.…”

“…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.…”