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

Stephenson, Philip Charles Robertson (1992) Subgroups of some (2, 3, n) triangle groups. PhD thesis, University of Glasgow.

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## Abstract

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  and ). 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.

Item Type: Thesis (PhD) Doctoral Q Science > QA Mathematics College of Science and Engineering > School of Mathematics and Statistics > Mathematics Stothers, Dr. W.W. 1992 Mrs Marie Cairney glathesis:1992-3054 Copyright of this thesis is held by the author. 07 Dec 2011 05 Feb 2014 11:03 http://theses.gla.ac.uk/id/eprint/3054 View Item