Categories for finite/permutation/symmetric groups

[nthiery]

This patch:

• Introduces two new categories: FiniteGroups and FinitePermutationGroups

• As a result, this standardizes the interface of those groups
  (cardinality, one, ...).

• Puts all pari, permutation, and matrix groups in the corresponding
  categories. There remains to handle Galois groups in
  sage/rings/number_field/.

• Deprecates the abstract class sage.groups.group.FiniteGroup.
  Content moved to the FiniteGroups category (see cayley_graph).
  It is not used anymore anywhere in Sage's library.

• Merges cayley_graph with that for FiniteSemigroups:

  • Generalization to any Semigroups with an elements option
    (should this be vertices?) to handle large/infinite semigroups
  • The call:
    sage: G.cayley_graph(connecting_set = [a,b,c])
    is deprecated in favor of:
    sage: G.cayley_graph(generators = [a,b,c])
  • The following feature is removed:
    sage: G.cayley_graph(connecting_set = a)
  • side = "right" is now the default (was "twosided" for semigroups).
  • Removed forcing implementation = "networkx" in the produced graph.
    Note: this changed the order of the edges, which required fixing
    a test in sage.graphs.generic_graphs (color_by_label)

• Adds cool examples of Cayley graphs plots, courtesy of Sebastien Labbe

• Provides group_generators defined from gens, as well as
  semigroup_generators defined from group_generators in the finite
  and coxeter cases.

• Puts the SymmetricGroup in the FiniteWeylGroups category.
  Beware: as all Sage's permutation groups, this uses GAP's product
  convention coming from left-to-right composition of permutations,
  which can be surprising for combinatorists.
  Beware: the generators of SymmetricGroup(n) are now its canonical
  Weyl group generators, namely the elementary transpositions

• Adds an has_descent method to permutation group elements

• Makes all named permutation groups, as well as GL and SL have
  UniqueRepresentation,

• Makes more systematic use of TestSuite(...).run()

• Makes a minor improvement to FiniteEnumeratedSets tests for
  large finite enumerated sets

• Strips away some unused imports

• Updates a couple doctests here and there

Further debatable changes:

• The underlying set of an alternating or symmetric group is now a
  tuple. This is safer and helps UniqueRepresentation. However, this
  could break backward compatibility. Also, the repr is now of the
  form SymmetricGroup((1,3,4)) instead of SymmetricGroup([1,3,4]).

• Due to the switch to UniqueRepresentation, with:

   sage: F = GF(3); MS = MatrixSpace(F,2,2)
   sage: gens = [MS([[0,1],[1,0]]),MS([[1,1],[0,1]])]
   sage: G = MatrixGroup(gens)
   sage: H = GL(2,F)
  

  the following equality test fails:

   sage: H == G
  

  True

  Do we really want this feature? If yes, than the equality test
  inherited from UniqueRepresentation will have to be fixed.

CC: @sagetrac-sage-combinat vengoroso@gmail.com

Component: group theory

Keywords: Finite groups, permutation groups, symmetric groups

Author: Nicolas M. Thiéry

Reviewer: David Joyner, Javier López Peña

Merged: sage-4.3.3.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/8044

[nthiery]

comment:1

Non finalized patch on: http://combinat.sagemath.org/hgwebdir.cgi/patches/file/tip/trac_8044-categories_finite_groups-nt.patch

[nthiery]

Description changed:

--- 
+++ 
@@ -12,3 +12,4 @@
 - Makes a minor improvement to FiniteEnumeratedSets tests for
   large finite enumerated sets
 
+Current (to be finalized patch) on: http://combinat.sagemath.org/hgwebdir.cgi/patches/file/tip/trac_8044-categories_finite_groups-nt.patch. Prereviews welcome!

[nthiery]

Description changed:

--- 
+++ 
@@ -1,15 +1,48 @@
 This patch:
 
-- Introduces two new categories, FiniteGroups and FinitePermutationGroups
-- Deprecates the class sage.groups.group.FiniteGroup
-  (content moved to the FiniteGroups category; this is essentially
-  the cayley_graph method)
-- Puts all permutation groups and most other finite groups in the
-  corresponding categories
+- Introduces two new categories: FiniteGroups and FinitePermutationGroups
 - As a result, this standardizes the interface of those groups
-  (cardinality, one, ...), and reveals lots of failures in the
-  completely undocumented PariGroup (category not set properly yet)
+  (cardinality, one, ...).
+
+- Puts all permutation groups and some other finite groups in the
+  corresponding categories. There remains to handle finite matrix
+  groups and Galois groups in sage/rings/number_field/.
+
+- Deprecates the class sage.groups.group.FiniteGroup. Content moved
+  to the FiniteGroups category; this was essentially the cayley_graph
+
+- Merges cayley_graph with that for FiniteSemigroups. In the merging,
+  ``connecting_set`` was deprecated to ``generators``. Also providing
+  a single element by itself as connecting set is no more supported.
+  Finally the default is side = "right" (was "twosided" for
+  semigroups). This method is also generalized to Semigroups with an
+  ``elements`` option (should this be vertices?).
+
+  Why do we care about the produced graph using implementation = "networkx"?
+  I would tend to remove this implementation detail; this would
+  change the order of the edges, which requires fixing a test in
+  sage.graphs.generic_graphs
+
+- Provides group_generators defined from gens, as well as
+  semigroup_generators defined from group_generators in the finite
+  and coxeter cases.
+
+- Also puts the SymmetricGroup in the FiniteWeylGroups category.
+  Beware: unusual product convention
+  Beware: this changes the generators to the standard Weyl group
+  generators, that is the elementary transpositions
+- Adds an has_descent method to permutation group elements
+
+- Make all named permutation groups have UniqueRepresentation
+- Questionable: the underlying set of an alternating or symmetric
+  group is now a tuple. This is safer and helps UniqueRepresentation.
+  However, this could break backward compatibility.  Also, maybe we
+  want to keep the output as SymmetricGroup([1,3,4])?
+
+- Makes more systematic use of TestSuite(...).run()
 - Makes a minor improvement to FiniteEnumeratedSets tests for
   large finite enumerated sets
+- Strips away unused imports
+- Updates a couple doctests here and there
 
-Current (to be finalized patch) on: http://combinat.sagemath.org/hgwebdir.cgi/patches/file/tip/trac_8044-categories_finite_groups-nt.patch. Prereviews welcome!
+

[nthiery]

Changed keywords from Finite groups, permutation groups to Finite groups, permutation groups, symmetric groups

[nthiery]

comment:3

The attached patch passes all tests on 4.3.1 + the sage-combinat patches already merged in 4.3.2: !#7976 #7921 #7938 !#8028 #8001 !#5524

[nthiery]

Description changed:

--- 
+++ 
@@ -1,48 +1,26 @@
 This patch:
 
-- Introduces two new categories: FiniteGroups and FinitePermutationGroups
-- As a result, this standardizes the interface of those groups
-  (cardinality, one, ...).
+* Introduces two new categories: FiniteGroups and FinitePermutationGroups
 
-- Puts all permutation groups and some other finite groups in the
-  corresponding categories. There remains to handle finite matrix
-  groups and Galois groups in sage/rings/number_field/.
+* Puts all permutation groups and some other finite groups in the corresponding categories. There remains to handle finite matrix groups and Galois groups in sage/rings/number_field/.
 
-- Deprecates the class sage.groups.group.FiniteGroup. Content moved
-  to the FiniteGroups category; this was essentially the cayley_graph
+* As a result, this standardizes the interface of those groups (cardinality, one, ...).
 
-- Merges cayley_graph with that for FiniteSemigroups. In the merging,
-  ``connecting_set`` was deprecated to ``generators``. Also providing
-  a single element by itself as connecting set is no more supported.
-  Finally the default is side = "right" (was "twosided" for
-  semigroups). This method is also generalized to Semigroups with an
-  ``elements`` option (should this be vertices?).
+* Deprecates the class sage.groups.group.[Trac macro FiniteGroup](https://trac.sagemath.org/wiki/WikiMacros#FiniteGroup-macro). Content moved to the FiniteGroups category; this was essentially the cayley_graph
 
-  Why do we care about the produced graph using implementation = "networkx"?
-  I would tend to remove this implementation detail; this would
-  change the order of the edges, which requires fixing a test in
-  sage.graphs.generic_graphs
+* Merges cayley_graph with that for FiniteSemigroups. In the merging, connecting_set was deprecated to generators. Also providing a single element by itself as connecting set is no more supported. Finally the default is side = "right" (was "twosided" for semigroups). This method is also generalized to Semigroups with an elements option (should this be vertices?).
 
-- Provides group_generators defined from gens, as well as
-  semigroup_generators defined from group_generators in the finite
-  and coxeter cases.
+  Why do we care about the produced graph using implementation = "networkx"? I would tend to remove this implementation detail; this would change the order of the edges, which requires fixing a test in sage.graphs.generic_graphs
 
-- Also puts the SymmetricGroup in the FiniteWeylGroups category.
-  Beware: unusual product convention
-  Beware: this changes the generators to the standard Weyl group
-  generators, that is the elementary transpositions
-- Adds an has_descent method to permutation group elements
+* Provides group_generators defined from gens, as well as semigroup_generators defined from group_generators in the finite and coxeter cases.
 
-- Make all named permutation groups have UniqueRepresentation
-- Questionable: the underlying set of an alternating or symmetric
-  group is now a tuple. This is safer and helps UniqueRepresentation.
-  However, this could break backward compatibility.  Also, maybe we
-  want to keep the output as SymmetricGroup([1,3,4])?
+* Also puts the SymmetricGroup in the FiniteWeylGroups category. Beware: unusual product convention Beware: this changes the generators to the standard Weyl group generators, that is the elementary transpositions
+* Adds an has_descent method to permutation group elements
 
-- Makes more systematic use of TestSuite(...).run()
-- Makes a minor improvement to FiniteEnumeratedSets tests for
-  large finite enumerated sets
-- Strips away unused imports
-- Updates a couple doctests here and there
+* Make all named permutation groups have UniqueRepresentation
+* Questionable: the underlying set of an alternating or symmetric group is now a tuple. This is safer and helps UniqueRepresentation. However, this could break backward compatibility.  Also, maybe we want to keep the output as SymmetricGroup([1,3,4])?
 
-
+* Makes more systematic use of TestSuite(...).run()
+* Makes a minor improvement to FiniteEnumeratedSets tests for large finite enumerated sets
+* Strips away unused imports
+* Updates a couple doctests here and there