sage.groups: Modularization fixes, # needs, doctest cosmetics

src/sage/combinat/root_system/coxeter_group.py

@@ -9,9 +9,11 @@
 #                  https://www.gnu.org/licenses/
 # ***************************************************************************
 
-from sage.combinat.root_system.weyl_group import WeylGroup
-from sage.combinat.root_system.reflection_group_real import ReflectionGroup
 from sage.combinat.root_system.cartan_type import CartanType
+from sage.misc.lazy_import import lazy_import
+
+lazy_import('from sage.combinat.root_system.reflection_group_real', 'ReflectionGroup')
+lazy_import('sage.combinat.root_system.weyl_group', 'WeylGroup')
 
 
 def CoxeterGroup(data, implementation="reflection", base_ring=None, index_set=None):

src/sage/groups/abelian_gps/abelian_group_element.py

@@ -97,16 +97,16 @@ def as_permutation(self):
 
         EXAMPLES::
 
-            sage: G = AbelianGroup(3,[2,3,4],names="abc"); G
+            sage: G = AbelianGroup(3, [2,3,4], names="abc"); G
             Multiplicative Abelian group isomorphic to C2 x C3 x C4
             sage: a,b,c = G.gens()
-            sage: Gp = G.permutation_group(); Gp
+            sage: Gp = G.permutation_group(); Gp                                        # needs sage.groups
             Permutation Group with generators [(6,7,8,9), (3,4,5), (1,2)]
-            sage: a.as_permutation()
+            sage: a.as_permutation()                                                    # needs sage.libs.gap
             (1,2)
-            sage: ap = a.as_permutation(); ap
+            sage: ap = a.as_permutation(); ap                                           # needs sage.libs.gap
             (1,2)
-            sage: ap in Gp
+            sage: ap in Gp                                                              # needs sage.groups sage.libs.gap
             True
         """
         from sage.libs.gap.libgap import libgap
@@ -138,13 +138,14 @@ def word_problem(self, words):
 
         EXAMPLES::
 
-            sage: G = AbelianGroup(2,[2,3], names="xy")
+            sage: # needs sage.libs.gap
+            sage: G = AbelianGroup(2, [2,3], names="xy")
             sage: x,y = G.gens()
             sage: x.word_problem([x,y])
             [[x, 1]]
             sage: y.word_problem([x,y])
             [[y, 1]]
-            sage: v = (y*x).word_problem([x,y]); v #random
+            sage: v = (y*x).word_problem([x,y]); v  # random
             [[x, 1], [y, 1]]
             sage: prod([x^i for x,i in v]) == y*x
             True

src/sage/groups/abelian_gps/dual_abelian_group.py

@@ -25,6 +25,7 @@
     sage: F = AbelianGroup(5, [2,5,7,8,9], names='abcde')
     sage: (a, b, c, d, e) = F.gens()
 
+    sage: # needs sage.rings.number_field
     sage: Fd = F.dual_group(names='ABCDE')
     sage: Fd.base_ring()
     Cyclotomic Field of order 2520 and degree 576
@@ -34,6 +35,7 @@
     sage: A(b), A(c), A(d), A(e)
     (1, 1, 1, 1)
 
+    sage: # needs sage.rings.real_mpfr
     sage: Fd = F.dual_group(names='ABCDE', base_ring=CC)
     sage: A,B,C,D,E = Fd.gens()
     sage: A(a)    # abs tol 1e-8
@@ -78,6 +80,7 @@ def is_DualAbelianGroup(x):
 
     EXAMPLES::
 
+        sage: # needs sage.rings.number_field
         sage: from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroup
         sage: F = AbelianGroup(5,[3,5,7,8,9], names=list("abcde"))
         sage: Fd = F.dual_group()
@@ -100,11 +103,12 @@ class DualAbelianGroup_class(UniqueRepresentation, AbelianGroupBase):
     EXAMPLES::
 
         sage: F = AbelianGroup(5,[3,5,7,8,9], names="abcde")
-        sage: F.dual_group()
+        sage: F.dual_group()                                                            # needs sage.rings.number_field
         Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
         over Cyclotomic Field of order 2520 and degree 576
+
         sage: F = AbelianGroup(4,[15,7,8,9], names="abcd")
-        sage: F.dual_group(base_ring=CC)
+        sage: F.dual_group(base_ring=CC)                                                # needs sage.rings.real_mpfr
         Dual of Abelian Group isomorphic to Z/15Z x Z/7Z x Z/8Z x Z/9Z
         over Complex Field with 53 bits of precision
     """
@@ -117,7 +121,7 @@ def __init__(self, G, names, base_ring):
         EXAMPLES::
 
             sage: F = AbelianGroup(5,[3,5,7,8,9], names="abcde")
-            sage: F.dual_group()
+            sage: F.dual_group()                                                        # needs sage.rings.number_field
             Dual of Abelian Group isomorphic to Z/3Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
             over Cyclotomic Field of order 2520 and degree 576
        """
@@ -174,12 +178,13 @@ def _repr_(self):
         EXAMPLES::
 
             sage: F = AbelianGroup(5, [2,5,7,8,9], names='abcde')
-            sage: Fd = F.dual_group(names='ABCDE', base_ring=CyclotomicField(2*5*7*8*9))
-            sage: Fd   # indirect doctest
+            sage: Fd = F.dual_group(names='ABCDE',                                      # needs sage.rings.number_field
+            ....:                   base_ring=CyclotomicField(2*5*7*8*9))
+            sage: Fd   # indirect doctest                                               # needs sage.rings.number_field
             Dual of Abelian Group isomorphic to Z/2Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
             over Cyclotomic Field of order 5040 and degree 1152
-            sage: Fd = F.dual_group(names='ABCDE', base_ring=CC)
-            sage: Fd
+            sage: Fd = F.dual_group(names='ABCDE', base_ring=CC)                        # needs sage.rings.real_mpfr
+            sage: Fd                                                                    # needs sage.rings.real_mpfr
             Dual of Abelian Group isomorphic to Z/2Z x Z/5Z x Z/7Z x Z/8Z x Z/9Z
             over Complex Field with 53 bits of precision
         """
@@ -202,8 +207,8 @@ def _latex_(self):
         EXAMPLES::
 
             sage: F = AbelianGroup(3, [2]*3)
-            sage: Fd = F.dual_group()
-            sage: Fd._latex_()
+            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
+            sage: Fd._latex_()                                                          # needs sage.rings.number_field
             '$\\mathrm{DualAbelianGroup}( AbelianGroup ( 3, (2, 2, 2) ) )$'
         """
         return r"$\mathrm{DualAbelianGroup}( AbelianGroup ( %s, %s ) )$" % (self.ngens(), self.gens_orders())
@@ -215,12 +220,13 @@ def random_element(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3,9])
-            sage: Gd = G.dual_group(base_ring=CC)
-            sage: Gd.random_element().parent() is Gd
+            sage: Gd = G.dual_group(base_ring=CC)                                       # needs sage.rings.real_mpfr
+            sage: Gd.random_element().parent() is Gd                                    # needs sage.rings.real_mpfr
             True
 
-            sage: N = 43^2-1
-            sage: G = AbelianGroup([N],names="a")
+            sage: # needs sage.rings.real_mpfr
+            sage: N = 43^2 - 1
+            sage: G = AbelianGroup([N], names="a")
             sage: Gd = G.dual_group(names="A", base_ring=CC)
             sage: a, = G.gens()
             sage: A, = Gd.gens()
@@ -243,7 +249,8 @@ def gen(self, i=0):
 
         EXAMPLES::
 
-            sage: F = AbelianGroup(3,[1,2,3],names='a')
+            sage: # needs sage.rings.number_field
+            sage: F = AbelianGroup(3, [1,2,3], names='a')
             sage: Fd = F.dual_group(names="A")
             sage: Fd.0
             1
@@ -270,8 +277,8 @@ def gens(self):
 
         EXAMPLES::
 
-            sage: F = AbelianGroup([7,11]).dual_group()
-            sage: F.gens()
+            sage: F = AbelianGroup([7,11]).dual_group()                                 # needs sage.rings.number_field
+            sage: F.gens()                                                              # needs sage.rings.number_field
             (X0, X1)
         """
         n = self.group().ngens()
@@ -284,8 +291,8 @@ def ngens(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([7]*100)
-            sage: Fd = F.dual_group()
-            sage: Fd.ngens()
+            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
+            sage: Fd.ngens()                                                            # needs sage.rings.number_field
             100
         """
         return self.group().ngens()
@@ -301,8 +308,8 @@ def gens_orders(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([5]*1000)
-            sage: Fd = F.dual_group()
-            sage: invs = Fd.gens_orders(); len(invs)
+            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
+            sage: invs = Fd.gens_orders(); len(invs)                                    # needs sage.rings.number_field
             1000
         """
         return self.group().gens_orders()
@@ -316,8 +323,8 @@ def invariants(self):
         EXAMPLES::
 
             sage: F = AbelianGroup([5]*1000)
-            sage: Fd = F.dual_group()
-            sage: invs = Fd.gens_orders(); len(invs)
+            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
+            sage: invs = Fd.gens_orders(); len(invs)                                    # needs sage.rings.number_field
             1000
         """
         # TODO: deprecate
@@ -331,9 +338,9 @@ def __contains__(self, X):
 
             sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names="abcde")
             sage: a,b,c,d,e = F.gens()
-            sage: Fd = F.dual_group(names = "ABCDE")
-            sage: A,B,C,D,E = Fd.gens()
-            sage: A*B^2*D^7 in Fd
+            sage: Fd = F.dual_group(names="ABCDE")                                      # needs sage.rings.number_field
+            sage: A,B,C,D,E = Fd.gens()                                                 # needs sage.rings.number_field
+            sage: A*B^2*D^7 in Fd                                                       # needs sage.rings.number_field
             True
         """
         return X.parent() == self and is_DualAbelianGroupElement(X)
@@ -345,8 +352,8 @@ def order(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3,9])
-            sage: Gd = G.dual_group()
-            sage: Gd.order()
+            sage: Gd = G.dual_group()                                                   # needs sage.rings.number_field
+            sage: Gd.order()                                                            # needs sage.rings.number_field
             54
         """
         G = self.group()
@@ -359,10 +366,10 @@ def is_commutative(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3,9])
-            sage: Gd = G.dual_group()
-            sage: Gd.is_commutative()
+            sage: Gd = G.dual_group()                                                   # needs sage.rings.number_field
+            sage: Gd.is_commutative()                                                   # needs sage.rings.number_field
             True
-            sage: Gd.is_abelian()
+            sage: Gd.is_abelian()                                                       # needs sage.rings.number_field
             True
         """
         return True
@@ -375,8 +382,8 @@ def list(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3], names="ab")
-            sage: Gd = G.dual_group(names="AB")
-            sage: Gd.list()
+            sage: Gd = G.dual_group(names="AB")                                         # needs sage.rings.number_field
+            sage: Gd.list()                                                             # needs sage.rings.number_field
             (1, B, B^2, A, A*B, A*B^2)
         """
         if not self.is_finite():
@@ -391,18 +398,21 @@ def __iter__(self):
         EXAMPLES::
 
             sage: G = AbelianGroup([2,3], names="ab")
-            sage: Gd = G.dual_group(names="AB")
-            sage: [X for X in Gd]
+            sage: Gd = G.dual_group(names="AB")                                         # needs sage.rings.number_field
+            sage: [X for X in Gd]                                                       # needs sage.rings.number_field
             [1, B, B^2, A, A*B, A*B^2]
-            sage: N = 43^2-1
-            sage: G = AbelianGroup([N],names="a")
+
+            sage: # needs sage.rings.real_mpfr
+            sage: N = 43^2 - 1
+            sage: G = AbelianGroup([N], names="a")
             sage: Gd = G.dual_group(names="A", base_ring=CC)
             sage: a, = G.gens()
             sage: A, = Gd.gens()
             sage: x = a^(N/4)
             sage: y = a^(N/3)
             sage: z = a^(N/14)
-            sage: len([X for X in Gd if abs(X(x)-1)>0.01 and abs(X(y)-1)>0.01 and abs(X(z)-1)>0.01])
+            sage: len([X for X in Gd
+            ....:      if abs(X(x)-1)>0.01 and abs(X(y)-1)>0.01 and abs(X(z)-1)>0.01])
             880
         """
         yield from self.list()

src/sage/groups/abelian_gps/dual_abelian_group_element.py

@@ -8,13 +8,13 @@
     sage: F
     Multiplicative Abelian group isomorphic to C2 x C3 x C5 x C7 x C8
 
-    sage: Fd = F.dual_group(names="ABCDE")
-    sage: Fd
+    sage: Fd = F.dual_group(names="ABCDE"); Fd                                          # needs sage.rings.number_field
     Dual of Abelian Group isomorphic to Z/2Z x Z/3Z x Z/5Z x Z/7Z x Z/8Z
     over Cyclotomic Field of order 840 and degree 192
 
 The elements of the dual group can be evaluated on elements of the original group::
 
+    sage: # needs sage.rings.number_field
     sage: a,b,c,d,e = F.gens()
     sage: A,B,C,D,E = Fd.gens()
     sage: A*B^2*D^7
@@ -71,10 +71,10 @@ def is_DualAbelianGroupElement(x) -> bool:
     EXAMPLES::
 
         sage: from sage.groups.abelian_gps.dual_abelian_group import is_DualAbelianGroupElement
-        sage: F = AbelianGroup(5, [5,5,7,8,9], names=list("abcde")).dual_group()
-        sage: is_DualAbelianGroupElement(F)
+        sage: F = AbelianGroup(5, [5,5,7,8,9], names=list("abcde")).dual_group()        # needs sage.rings.number_field
+        sage: is_DualAbelianGroupElement(F)                                             # needs sage.rings.number_field
         False
-        sage: is_DualAbelianGroupElement(F.an_element())
+        sage: is_DualAbelianGroupElement(F.an_element())                                # needs sage.rings.number_field
         True
     """
     return isinstance(x, DualAbelianGroupElement)
@@ -96,6 +96,7 @@ def __call__(self, g):
 
         EXAMPLES::
 
+            sage: # needs sage.rings.number_field
             sage: F = AbelianGroup(5, [2,3,5,7,8], names="abcde")
             sage: a,b,c,d,e = F.gens()
             sage: Fd = F.dual_group(names="ABCDE")
@@ -117,7 +118,7 @@ def __call__(self, g):
             sage: a, = F.gens()
             sage: Fd = F.dual_group(names="A", base_ring=GF(29))
             sage: A, = Fd.gens()
-            sage: A(a)
+            sage: A(a)                                                                  # needs sage.libs.pari
             16
         """
         F = self.parent().base_ring()
@@ -146,6 +147,7 @@ def word_problem(self, words):
 
         EXAMPLES::
 
+            sage: # needs sage.rings.number_field
             sage: G = AbelianGroup(5,[3, 5, 5, 7, 8], names="abcde")
             sage: Gd = G.dual_group(names="abcde")
             sage: a,b,c,d,e = Gd.gens()

src/sage/groups/abelian_gps/element_base.py

@@ -48,9 +48,9 @@ class AbelianGroupElementBase(MultiplicativeGroupElement):
     EXAMPLES::
 
         sage: F = AbelianGroup(3,[7,8,9])
-        sage: Fd = F.dual_group(names="ABC")
-        sage: A,B,C = Fd.gens()
-        sage: A*B^-1 in Fd
+        sage: Fd = F.dual_group(names="ABC")                                            # needs sage.rings.number_field
+        sage: A,B,C = Fd.gens()                                                         # needs sage.rings.number_field
+        sage: A*B^-1 in Fd                                                              # needs sage.rings.number_field
         True
     """
 
@@ -61,9 +61,9 @@ def __init__(self, parent, exponents):
         EXAMPLES::
 
             sage: F = AbelianGroup(3,[7,8,9])
-            sage: Fd = F.dual_group(names="ABC")
-            sage: A,B,C = Fd.gens()
-            sage: A*B^-1 in Fd
+            sage: Fd = F.dual_group(names="ABC")                                        # needs sage.rings.number_field
+            sage: A,B,C = Fd.gens()                                                     # needs sage.rings.number_field
+            sage: A*B^-1 in Fd                                                          # needs sage.rings.number_field
             True
 
         Check that :issue:`35216` is fixed::
@@ -125,7 +125,7 @@ def _libgap_(self):
         TESTS::
 
             sage: F.<a,b,c> = AbelianGroup([7,8,9])
-            sage: libgap(a**2 * c) * libgap(b * c**2)
+            sage: libgap(a**2 * c) * libgap(b * c**2)                                   # needs sage.libs.gap
             f1^2*f2*f6
         """
         from sage.misc.misc_c import prod
@@ -147,6 +147,7 @@ def list(self):
 
         EXAMPLES::
 
+            sage: # needs sage.rings.number_field
             sage: F = AbelianGroup(5,[2, 3, 5, 7, 8], names="abcde")
             sage: a,b,c,d,e = F.gens()
             sage: Ad = F.dual_group(names="ABCDE")
@@ -208,8 +209,8 @@ def _richcmp_(self, other, op):
             sage: a > b
             True
 
-            sage: Gd.<A,B> = G.dual_group()
-            sage: A > B
+            sage: Gd.<A,B> = G.dual_group()                                             # needs sage.rings.number_field
+            sage: A > B                                                                 # needs sage.rings.number_field
             True
         """
         return richcmp(self._exponents, other._exponents, op)
@@ -226,9 +227,9 @@ def order(self):
         EXAMPLES::
 
             sage: F = AbelianGroup(3,[7,8,9])
-            sage: Fd = F.dual_group()
-            sage: A,B,C = Fd.gens()
-            sage: (B*C).order()
+            sage: Fd = F.dual_group()                                                   # needs sage.rings.number_field
+            sage: A,B,C = Fd.gens()                                                     # needs sage.rings.number_field
+            sage: (B*C).order()                                                         # needs sage.rings.number_field
             72
 
             sage: F = AbelianGroup(3,[7,8,9]); F