Subgroups of some (2, 3, n) triangle groups

As an abstract group, the (2,3,n) triangle group has the presentation mit _n = < x,y : x^2 = y^3 = (yx)^n = 1 > This thesis is concerned with subgroups of finite index in mit 9, mit _11 and mit 13. With a subgroup of finite index, u, in the (2,3,11) triangle group, we associate a quintuple of non-negative integers (u,p,e,f,g), with u 1 and 5u = 132(p - 1) + 33e + 44f + 60g. We show in Theorem 1.4.6 that each quintuple, satisfying the conditions, corresponds to a subgroup of mit 11. With a subgroup of finite index, u, in the (2,3,12) triangle group, we associate a quintuple of non-negative integers (u,p,e,f,g), with u 1 and 7u = 156(p - 1) + 39e + 52f + 72g. We show in Theorem 3.3.6 that each quintuple, satisfying the conditions, corresponds to a subgroup of mit 13. With a subgroup of finite index, u, in the (2,3,9) triangle group, we associate a sextuple of non-negative integers (u,p,e,f,g1,g3) with u 1, u = f (mod 3) and u = 36(p - 1) + 9e + 12f + 16g_1 + 12g_3. We show in Theorem 2,3,9 that each sextuple, satisfying the conditions, corresponds to a subgroup of mit 9 with the following exceptions: (a) (12n+ 9,0,1,0,0,n+ 3), V n 0 (b) (24,0,0,0,0,5) (c) (24,0,0,0,3,1) (d) (24,0,0,3,0,2) Coset diagrams are used extensively in the proofs, although to prove exception (a) for mit 9, we make use of Hauptmodul equations (see [1] and [23]). Computer programs were developed to generate all quintuples satisfying the relevant conditions for (2,3,110 subgroups for u 101, all quintuples satisfying the relevant conditions for (2,3,13) subgroups for u 110, and all sextuples satisfying the relevant conditions for (2,3,9) subgroups for u 38. These programs and their output are presented in the Appendices. We show in Theorem 1.2.2 that quintuples, which satisfy the relevant (2,3,11) conditions, exist for each u 99. We show in Theorem 2.2.1 that sextuples, which satisfy the relevant (2,3,9) conditions, exist for each u 36. We show in Theorem 3.2.1 that quintuples, which satisfy the relevant (2,3,13) conditions, exist for each u 104.