Coverage for local/lib/python2.7/site-packages/sage/groups/groups_catalog.py : 100%

r""" 

Examples of Groups 

The ``groups`` object may be used to access examples of various groups. 

Using tab-completion on this object is an easy way to discover and quickly 

create the groups that are available (as listed here). 

Let ``<tab>`` indicate pressing the tab key. So begin by typing 

``groups.<tab>`` to the see primary divisions, followed by (for example) 

``groups.matrix.<tab>`` to access various groups implemented as sets of matrices. 

- Permutation Groups (``groups.permutation.<tab>``) 

- :class:`groups.permutation.Symmetric <sage.groups.perm_gps.permgroup_named.SymmetricGroup>` 

- :class:`groups.permutation.Alternating <sage.groups.perm_gps.permgroup_named.AlternatingGroup>` 

- :class:`groups.permutation.KleinFour <sage.groups.perm_gps.permgroup_named.KleinFourGroup>` 

- :class:`groups.permutation.Quaternion <sage.groups.perm_gps.permgroup_named.QuaternionGroup>` 

- :class:`groups.permutation.Cyclic <sage.groups.perm_gps.permgroup_named.CyclicPermutationGroup>` 

- :class:`groups.permutation.Dihedral <sage.groups.perm_gps.permgroup_named.DihedralGroup>` 

- :class:`groups.permutation.DiCyclic <sage.groups.perm_gps.permgroup_named.DiCyclicGroup>` 

- :class:`groups.permutation.Mathieu <sage.groups.perm_gps.permgroup_named.MathieuGroup>` 

- :class:`groups.permutation.Suzuki <sage.groups.perm_gps.permgroup_named.SuzukiGroup>` 

- :class:`groups.permutation.PGL <sage.groups.perm_gps.permgroup_named.PGL>` 

- :class:`groups.permutation.PSL <sage.groups.perm_gps.permgroup_named.PSL>` 

- :class:`groups.permutation.PSp <sage.groups.perm_gps.permgroup_named.PSp>` 

- :class:`groups.permutation.PSU <sage.groups.perm_gps.permgroup_named.PSU>` 

- :class:`groups.permutation.PGU <sage.groups.perm_gps.permgroup_named.PGU>` 

- :class:`groups.permutation.Transitive <sage.groups.perm_gps.permgroup_named.TransitiveGroup>` 

- :class:`groups.permutation.RubiksCube <sage.groups.perm_gps.cubegroup.CubeGroup>` 

- Matrix Groups (``groups.matrix.<tab>``) 

- :func:`groups.matrix.QuaternionGF3 <sage.groups.matrix_gps.finitely_generated.QuaternionMatrixGroupGF3>` 

- :func:`groups.matrix.GL <sage.groups.matrix_gps.linear.GL>` 

- :func:`groups.matrix.SL <sage.groups.matrix_gps.linear.SL>` 

- :func:`groups.matrix.Sp <sage.groups.matrix_gps.symplectic.Sp>` 

- :func:`groups.matrix.GU <sage.groups.matrix_gps.unitary.GU>` 

- :func:`groups.matrix.SU <sage.groups.matrix_gps.unitary.SU>` 

- :func:`groups.matrix.GO <sage.groups.matrix_gps.orthogonal.GO>` 

- :func:`groups.matrix.SO <sage.groups.matrix_gps.orthogonal.SO>` 

- Finitely Presented Groups (``groups.presentation.<tab>``) 

- :func:`groups.presentation.Alternating <sage.groups.finitely_presented_named.AlternatingPresentation>` 

- :func:`groups.presentation.Cyclic <sage.groups.finitely_presented_named.CyclicPresentation>` 

- :func:`groups.presentation.Dihedral <sage.groups.finitely_presented_named.DihedralPresentation>` 

- :func:`groups.presentation.DiCyclic <sage.groups.finitely_presented_named.DiCyclicPresentation>` 

- :func:`groups.presentation.FGAbelian <sage.groups.finitely_presented_named.FinitelyGeneratedAbelianPresentation>` 

- :func:`groups.presentation.KleinFour <sage.groups.finitely_presented_named.KleinFourPresentation>` 

- :func:`groups.presentation.Quaternion <sage.groups.finitely_presented_named.QuaternionPresentation>` 

- :func:`groups.presentation.Symmetric <sage.groups.finitely_presented_named.SymmetricPresentation>` 

- Affine Groups (``groups.affine.<tab>``) 

- :func:`groups.affine.Affine <sage.groups.affine_gps.affine_group.AffineGroup>` 

- :func:`groups.affine.Euclidean <sage.groups.affine_gps.euclidean_group.EuclideanGroup>` 

- Miscellaneous Groups (``groups.misc.<tab>``) 

- Coxeter, reflection and related groups 

- :func:`groups.misc.Braid <sage.groups.braid.BraidGroup>` 

- :func:`groups.misc.CoxeterGroup <sage.combinat.root_system.coxeter_group.CoxeterGroup>` 

- :func:`groups.misc.ReflectionGroup <sage.combinat.root_system.reflection_group_real.ReflectionGroup>` 

- :class:`groups.misc.RightAngledArtin <sage.groups.raag.RightAngledArtinGroup>` 

- :func:`groups.misc.WeylGroup <sage.combinat.root_system.weyl_group.WeylGroup>` 

- other miscellanous groups 

- :func:`groups.misc.AdditiveAbelian <sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroup>` 

- :class:`groups.misc.AdditiveCyclic <sage.rings.finite_rings.integer_mod_ring.IntegerModFactory>` 

- :func:`groups.misc.Free <sage.groups.free_group.FreeGroup>` 

- :func:`groups.misc.SemimonomialTransformation <sage.groups.semimonomial_transformations.semimonomial_transformation_group.SemimonomialTransformationGroup>` 

""" 

# Implementation notes: 

# 

# With this groups_catalog.py module imported as 

# "groups" in all.py then groups.<tab> is available 

# 

# "catalog" modules are made available 

# as groups.matrix, etc by imports below 

# 

# Do not use this file for code 

# 

# Please keep this top-level clean, use 

# groups.misc for one-off examples 

# 

# Candidates for new primary divisions: 

# groups.sporadic - 26 sporadic groups 

# groups.misc - one-off stuff (implemented, but empty) 

# groups.presentation - free groups with relations 

# groups.symmetries - permutation groups of regular solids, or similar 

from sage.groups.matrix_gps import catalog as matrix 

from sage.groups.perm_gps import permutation_groups_catalog as permutation 

from sage.groups.misc_gps import misc_groups_catalog as misc 

from sage.groups.affine_gps import catalog as affine 

from sage.groups import finitely_presented_catalog as presentation