# Ticket #11598: trac_11598-docfixes.patch

File trac_11598-docfixes.patch, 2.4 KB (added by davidloeffler, 23 months ago)

Apply over trac_11598-all.patch

• ## sage/modular/arithgroup/arithgroup_perm.py

# HG changeset patch
# User David Loeffler <d.loeffler.01@cantab.net>
# Date 1310886741 -3600
# Node ID da17358e56688cd46a982c50f747c4aca57b705e
# Parent  f2839cfc46bf2c42a7675f28b439b61308230c34
#11598: documentation fixes

diff -r f2839cfc46bf -r da17358e5668 sage/modular/arithgroup/arithgroup_perm.py
 a Arithmetic subgroups defined by permutations of cosets A subgroup of finite index H of a finitely generated group G is completely described by the action of the generators of G on the right cosets H described by the action of a set of generators of G on the right cosets H \backslash G = \{Hg\}_{g \in G}. After some arbitrary choice of numbering one can identify the action of generators as elements of a symmetric group acting transitively (and satisfying the relations of the relators in G). As {\rm s_2^2 = s_3^3 = -1, \quad r = s_2^{-1}\ l^{-1}\ s_2. In particular not all four are needed to generate the whole group {\rm SL}_2(\ZZ). Three couples which which generate {\rm SL}_2(\ZZ) are of particular interest: - (l,r) as the pair is involved in the continued fraction algorithm, - (l,s_2) similar as the one above because of the relations, SL}_2(\ZZ). Three couples which generate {\rm SL}_2(\ZZ) are of particular interest: - (l,r) as they are also semigroup generators for the semigroup of matrices in {\rm SL}_2(\ZZ) with non-negative entries, - (l,s_2) as they are closely related to the continued fraction algorithm, - (s_2,s_3) as the group {\rm PSL}_2(\ZZ) is the free product of the finite cyclic groups generated by these two elements. Part of these functions are based on Chris Kurth's *KFarey* package [Kur08]_. For tests see the file :mod:sage.modular.arithgroup.tests. For tests see the file sage.modular.arithgroup.tests. REFERENCES: A subgroup of {\rm SL}_2(\ZZ) defined by the action of generators on its coset graph. The class stores the action of generators s_2,s_3,l,r on right cosets The class stores the action of generators s_2, s_3, l, r on right cosets Hg of a finite index subgroup H < {\rm SL}_2(\ZZ). In particular the action of {\rm SL}_2(\ZZ)` on the cosets is on right.