gh-37128: Save and load finitely presented groups coming from libgap groups
    
<!-- Describe your changes here in detail -->
At this point it is not possible to load a saved finitely presented
group that comes from a libgap group, see an example in
#37061

One possible cause is the use of a general `__reduce__` method for free
groups. At least, adding such a method allows to load free groups or
finitely presented groups obtained from a libgap group using
`wrapFreeGroup` or `wrapFpGroup`. It fixes #37061 and it would simplify
some code in #36768

With these changes, free and finitely presented groups, included libgap
groups, can be pickled.

### 📝 Checklist



- [X] The title is concise, informative, and self-explanatory.
- [X] The description explains in detail what this PR is about.
- [X] I have linked a relevant issue or discussion.
- [X] I have created tests covering the changes.
- [X] I have updated the documentation accordingly.
    
URL: #37128
Reported by: Enrique Manuel Artal Bartolo
Reviewer(s): Enrique Manuel Artal Bartolo, Travis Scrimshaw

Author: Release Manager

src/sage/groups/artin.py

@@ -21,13 +21,14 @@
 #                  http://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.misc.cachefunc import cached_method
-from sage.groups.free_group import FreeGroup
-from sage.groups.finitely_presented import FinitelyPresentedGroup, FinitelyPresentedGroupElement
 from sage.combinat.root_system.coxeter_matrix import CoxeterMatrix
 from sage.combinat.root_system.coxeter_group import CoxeterGroup
+from sage.groups.free_group import FreeGroup
+from sage.groups.finitely_presented import FinitelyPresentedGroup, FinitelyPresentedGroupElement
+from sage.misc.cachefunc import cached_method
 from sage.rings.infinity import Infinity
 from sage.structure.richcmp import richcmp, rich_to_bool
+from sage.structure.unique_representation import UniqueRepresentation
 
 
 class ArtinGroupElement(FinitelyPresentedGroupElement):
@@ -329,7 +330,7 @@ def _left_normal_form_coxeter(self):
         return tuple([-delta] + form)
 
 
-class ArtinGroup(FinitelyPresentedGroup):
+class ArtinGroup(UniqueRepresentation, FinitelyPresentedGroup):
     r"""
     An Artin group.
 

src/sage/groups/cubic_braid.py

@@ -91,6 +91,8 @@
 from sage.groups.braid import BraidGroup
 from sage.misc.cachefunc import cached_method
 from sage.rings.integer import Integer
+from sage.structure.unique_representation import UniqueRepresentation
+
 
 try:
     from sage.libs.gap.element import GapElement
@@ -565,7 +567,7 @@ def conv2domain(laur_pol):
 #                  Class CubicBraidGroup
 #
 ##############################################################################
-class CubicBraidGroup(FinitelyPresentedGroup):
+class CubicBraidGroup(UniqueRepresentation, FinitelyPresentedGroup):
     r"""
     Factor groups of the Artin braid group mapping their generators to elements
     of order 3.
@@ -750,7 +752,6 @@ def __init__(self, names, cbg_type=None):
         n = Integer(len(names))
         if n < 1:
             raise ValueError("the number of strands must be an integer larger than one")
-
         if cbg_type is None:
             cbg_type = CubicBraidGroup.type.Coxeter
         if not isinstance(cbg_type, CubicBraidGroup.type):

src/sage/groups/finitely_presented.py

@@ -130,7 +130,6 @@
 
 from sage.arith.misc import GCD as gcd
 from sage.categories.morphism import SetMorphism
-from sage.functions.generalized import sign
 from sage.groups.free_group import FreeGroup
 from sage.groups.free_group import FreeGroupElement
 from sage.groups.group import Group
@@ -143,7 +142,8 @@
 from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
 from sage.rings.rational_field import QQ
 from sage.sets.set import Set
-from sage.structure.unique_representation import UniqueRepresentation
+from sage.structure.richcmp import richcmp, richcmp_method
+from sage.structure.unique_representation import CachedRepresentation
 
 
 class GroupMorphismWithGensImages(SetMorphism):
@@ -362,51 +362,7 @@ def __call__(self, *values, **kwds):
         return super().__call__(values)
 
 
-def wrap_FpGroup(libgap_fpgroup):
-    """
-    Wrap a GAP finitely presented group.
-
-    This function changes the comparison method of
-    ``libgap_free_group`` to comparison by Python ``id``. If you want
-    to put the LibGAP free group into a container ``(set, dict)`` then you
-    should understand the implications of
-    :meth:`~sage.libs.gap.element.GapElement._set_compare_by_id`. To
-    be safe, it is recommended that you just work with the resulting
-    Sage :class:`FinitelyPresentedGroup`.
-
-    INPUT:
-
-    - ``libgap_fpgroup`` -- a LibGAP finitely presented group
-
-    OUTPUT: a Sage :class:`FinitelyPresentedGroup`
-
-    EXAMPLES:
-
-    First construct a LibGAP finitely presented group::
-
-        sage: F = libgap.FreeGroup(['a', 'b'])
-        sage: a_cubed = F.GeneratorsOfGroup()[0] ^ 3
-        sage: P = F / libgap([ a_cubed ]);   P
-        <fp group of size infinity on the generators [ a, b ]>
-        sage: type(P)
-        <class 'sage.libs.gap.element.GapElement'>
-
-    Now wrap it::
-
-        sage: from sage.groups.finitely_presented import wrap_FpGroup
-        sage: wrap_FpGroup(P)
-        Finitely presented group < a, b | a^3 >
-    """
-    assert libgap_fpgroup.IsFpGroup()
-    libgap_fpgroup._set_compare_by_id()
-    from sage.groups.free_group import wrap_FreeGroup
-    free_group = wrap_FreeGroup(libgap_fpgroup.FreeGroupOfFpGroup())
-    relations = tuple(free_group(rel.UnderlyingElement())
-                      for rel in libgap_fpgroup.RelatorsOfFpGroup())
-    return FinitelyPresentedGroup(free_group, relations)
-
-
-class RewritingSystem:
+class RewritingSystem():
     """
     A class that wraps GAP's rewriting systems.
 
@@ -730,7 +686,8 @@ def make_confluent(self):
             raise ValueError('could not make the system confluent')
 
 
-class FinitelyPresentedGroup(GroupMixinLibGAP, UniqueRepresentation, Group, ParentLibGAP):
+@richcmp_method
+class FinitelyPresentedGroup(GroupMixinLibGAP, CachedRepresentation, Group, ParentLibGAP):
     """
     A class that wraps GAP's Finitely Presented Groups.
 
@@ -739,7 +696,9 @@ class FinitelyPresentedGroup(GroupMixinLibGAP, UniqueRepresentation, Group, Pare
         You should use
         :meth:`~sage.groups.free_group.FreeGroup_class.quotient` to
         construct finitely presented groups as quotients of free
-        groups.
+        groups. Any class inheriting this one should define
+        ``__reduce__ = CachedRepresentation.__reduce__``
+        after importing ``CachedRepresentation``.
 
     EXAMPLES::
 
@@ -770,7 +729,7 @@ class FinitelyPresentedGroup(GroupMixinLibGAP, UniqueRepresentation, Group, Pare
     """
     Element = FinitelyPresentedGroupElement
 
-    def __init__(self, free_group, relations, category=None):
+    def __init__(self, free_group, relations, category=None, libgap_fpgroup=None):
         """
         The Python constructor.
 
@@ -786,6 +745,25 @@ def __init__(self, free_group, relations, category=None):
             sage: J is H
             True
 
+            sage: A5 = libgap(AlternatingGroup(5))
+            sage: A5gapfp = A5.IsomorphismFpGroup().Range()
+            sage: A5gapfp
+            <fp group of size 60 on the generators [ A_5.1, A_5.2 ]>
+            sage: A5sage = A5gapfp.sage(); A5sage;
+            Finitely presented group < A_5.1, A_5.2 | A_5.1^5*A_5.2^-5, A_5.1^5*(A_5.2^-1*A_5.1^-1)^2, (A_5.1^-2*A_5.2^2)^2 >
+            sage: A5sage.inject_variables()
+            Traceback (most recent call last):
+            ...
+            ValueError: variable names have not yet been set using self._assign_names(...)
+
+        Check that pickling works::
+
+            sage: G = FreeGroup(2) / [2 * (1, 2, -1, -2)]
+            sage: loads(dumps(G))
+            Finitely presented group < x0, x1 | (x0*x1*x0^-1*x1^-1)^2 >
+            sage: G.__reduce__()[1][1]
+            (Free Group on generators {x0, x1}, ((x0*x1*x0^-1*x1^-1)^2,))
+
             sage: TestSuite(H).run()
             sage: TestSuite(J).run()
         """
@@ -794,11 +772,47 @@ def __init__(self, free_group, relations, category=None):
         assert isinstance(relations, tuple)
         self._free_group = free_group
         self._relations = relations
-        self._assign_names(free_group.variable_names())
-        parent_gap = free_group.gap() / libgap([rel.gap() for rel in relations])
-        ParentLibGAP.__init__(self, parent_gap)
+        try:
+            self._assign_names(free_group.variable_names())
+        except ValueError:
+            pass
+        if libgap_fpgroup is None:
+            libgap_fpgroup = free_group.gap() / libgap([rel.gap() for rel in relations])
+        ParentLibGAP.__init__(self, libgap_fpgroup)
         Group.__init__(self, category=category)
 
+    def __hash__(self):
+        """
+        Make hashable.
+
+        EXAMPLES::
+
+            sage: G = FreeGroup(2) / [(1, 2, 2, 1)]
+            sage: G.__hash__() == hash((G.free_group(), G.relations()))
+            True
+        """
+        return hash((self._free_group, self._relations))
+
+    def __richcmp__(self, other, op):
+        """
+        Rich comparison of ``self`` and ``other``.
+
+        EXAMPLES::
+
+            sage: G1 = FreeGroup(2) / [(1, 2, 2, 1, 2, 1)]
+            sage: G2 = libgap(G1).sage()
+            sage: G1 == G2
+            True
+            sage: G1 is G2
+            False
+        """
+        if not isinstance(other, self.__class__):
+            from sage.structure.richcmp import op_NE
+            return (op == op_NE)
+        self_data = (self._free_group, self._relations)
+        other_data = (other._free_group, other._relations)
+        return richcmp(self_data, other_data, op)
+
     def _repr_(self) -> str:
         """
         Return a string representation.
@@ -812,7 +826,7 @@ def _repr_(self) -> str:
             sage: H._repr_()
             'Finitely presented group < a, b | a, b^3 >'
         """
-        gens = ', '.join(self.variable_names())
+        gens = ', '.join(self._free_group._gen_names)
         rels = ', '.join(str(r) for r in self.relations())
         return 'Finitely presented group ' + '< ' + gens + ' | ' + rels + ' >'
 
@@ -1357,15 +1371,15 @@ def abelianization_map(self):
         hom_ab_fp = ab_libgap.IsomorphismFpGroup()
         ab_libgap_fp = hom_ab_fp.Range()
         hom_simply = ab_libgap_fp.IsomorphismSimplifiedFpGroup()
-        ab = wrap_FpGroup(hom_simply.Range())
+        ab = hom_simply.Range().sage()
         images = []
         for f in self.gens():
             f0 = hom_ab_libgap.Image(f)
             f1 = hom_ab_fp.Image(f0)
             f2 = hom_simply.Image(f1)
             L = f2.UnderlyingElement().LetterRepAssocWord()
             images.append(ab([int(j) for j in L]))
-        return self.hom(codomain=ab, im_gens=images)
+        return self.hom(codomain=ab, im_gens=images, check=False)
 
     @cached_method
     def abelianization_to_algebra(self, ring=QQ):
@@ -1438,12 +1452,9 @@ def simplification_isomorphism(self):
             sage: H = G / [a*b*c, a*b^2, c*b/c^2]
             sage: I = H.simplification_isomorphism()
             sage: I
-            Generic morphism:
+            Group morphism:
               From: Finitely presented group < a, b, c | a*b*c, a*b^2, c*b*c^-2 >
               To:   Finitely presented group < b |  >
-              Defn: a |--> b^-2
-                    b |--> b
-                    c |--> b
             sage: I(a)
             b^-2
             sage: I(b)
@@ -1455,21 +1466,26 @@ def simplification_isomorphism(self):
 
             sage: F = FreeGroup(1)
             sage: G = F.quotient([F.0])
-            sage: G.simplification_isomorphism()
-            Generic morphism:
+            sage: h = G.simplification_isomorphism(); h
+            Group morphism:
               From: Finitely presented group < x | x >
               To:   Finitely presented group <  |  >
-              Defn: x |--> 1
+            sage: h(G.gen(0))
+            1
 
         ALGORITHM:
 
         Uses GAP.
         """
         II = self.gap().IsomorphismSimplifiedFpGroup()
-        codomain = wrap_FpGroup(II.Range())
-        phi = lambda x: codomain(II.ImageElm(x.gap()))
-        HS = self.Hom(codomain)
-        return GroupMorphismWithGensImages(HS, phi)
+        cod = II.Range().sage()
+        phi = [cod(II.ImageElm(x)) for x in self.gap().GeneratorsOfGroup()]
+        return self.hom(codomain=cod, im_gens=phi, check=False)
+        # II = self.gap().IsomorphismSimplifiedFpGroup()
+        # codomain = II.Range().sage()
+        # phi = lambda x: codomain(II.ImageElm(x.gap()))
+        # HS = self.Hom(codomain)
+        # return GroupMorphismWithGensImages(HS, phi)
 
     def simplified(self):
         """
@@ -1543,48 +1559,44 @@ def epimorphisms(self, H):
             sage: F = FreeGroup(3)
             sage: G = F / [F([1, 2, 3, 1, 2, 3]), F([1, 1, 1])]
             sage: H = AlternatingGroup(3)
-            sage: G.epimorphisms(H)
-            [Generic morphism:
-               From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
-               To:   Alternating group of order 3!/2 as a permutation group
-               Defn: x0 |--> ()
-                     x1 |--> (1,3,2)
-                     x2 |--> (1,2,3),
-             Generic morphism:
-               From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
-               To:   Alternating group of order 3!/2 as a permutation group
-               Defn: x0 |--> (1,3,2)
-                     x1 |--> ()
-                     x2 |--> (1,2,3),
-             Generic morphism:
-               From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
-               To:   Alternating group of order 3!/2 as a permutation group
-               Defn: x0 |--> (1,3,2)
-                     x1 |--> (1,2,3)
-                     x2 |--> (),
-             Generic morphism:
-               From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
-               To:   Alternating group of order 3!/2 as a permutation group
-               Defn: x0 |--> (1,2,3)
-                     x1 |--> (1,2,3)
-                     x2 |--> (1,2,3)]
+            sage: for quo in G.epimorphisms(H):
+            ....:   for a in G.gens():
+            ....:       print(a, "|-->", quo(a))
+            ....:   print("-----")
+            x0 |--> ()
+            x1 |--> (1,3,2)
+            x2 |--> (1,2,3)
+            -----
+            x0 |--> (1,3,2)
+            x1 |--> ()
+            x2 |--> (1,2,3)
+            -----
+            x0 |--> (1,3,2)
+            x1 |--> (1,2,3)
+            x2 |--> ()
+            -----
+            x0 |--> (1,2,3)
+            x1 |--> (1,2,3)
+            x2 |--> (1,2,3)
+            -----
 
         ALGORITHM:
 
         Uses libgap's GQuotients function.
         """
-        from sage.misc.misc_c import prod
-        HomSpace = self.Hom(H)
+        # from sage.misc.misc_c import prod
+        # HomSpace = self.Hom(H)
         Gg = libgap(self)
         Hg = libgap(H)
         gquotients = Gg.GQuotients(Hg)
         res = []
         # the following closure is needed to attach a specific value of quo to
         # each function in the different morphisms
-        fmap = lambda tup: (lambda a: H(prod(tup[abs(i)-1]**sign(i) for i in a.Tietze())))
+        # fmap = lambda tup: (lambda a: H(prod(tup[abs(i)-1]**sign(i) for i in a.Tietze())))
         for quo in gquotients:
-            tup = tuple(H(quo.ImageElm(i.gap()).sage()) for i in self.gens())
-            fhom = GroupMorphismWithGensImages(HomSpace, fmap(tup))
+            # tup = tuple(H(quo.ImageElm(i.gap()).sage()) for i in self.gens())
+            # fhom = GroupMorphismWithGensImages(HomSpace, fmap(tup))
+            fhom = self.hom(codomain=H, im_gens=[H(quo.ImageElm(a.gap())) for a in self.gens()])
             res.append(fhom)
         return res