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sagemath
GitHub Repository: sagemath/sagesmc
 Path: blob/master/src/sage/groups/groups_catalog.py
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r"""

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Examples of Groups

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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"""

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# Implementation notes:

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#

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#   With this  groups_catalog.py  module imported as

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

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#

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#   "catalog" modules are made available

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#   as groups.matrix, etc by imports below

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#

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#   Do not use this file for code

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#

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#   Please keep this top-level clean, use

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#   groups.misc for one-off examples

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#

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#   Candidates for new primary divisions:

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#       groups.sporadic - 26 sporadic groups

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

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#       groups.presentation - free groups with relations

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

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from sage.groups.matrix_gps import catalog as matrix

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from sage.groups.perm_gps import permutation_groups_catalog as permutation

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from sage.groups.misc_gps import misc_groups_catalog as misc

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from sage.groups.affine_gps import catalog as affine

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from sage.groups import finitely_presented_catalog as presentation

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