Fixing doctests due to bad rebase and some doc tweaks.

src/sage/algebras/group_algebra.py

@@ -63,15 +63,15 @@ def GroupAlgebra(G, R=IntegerRing()):
         sage: A = GroupAlgebra(G, R); A
         Algebra of Dihedral group of order 6 as a permutation group over Rational Field
         sage: a = A.an_element(); a
-        () + (1,2) + 3*(1,2,3) + 2*(1,3)
+        () + (1,2) + 3*(1,2,3) + 2*(1,3,2)
 
     This space is endowed with an algebra structure, obtained by extending
     by bilinearity the multiplication of `G` to a multiplication on `RG`::
 
         sage: A in Algebras
         True
         sage: a * a
-        6*() + 9*(2,3) + 8*(1,2) + 8*(1,2,3) + 11*(1,3,2) + 7*(1,3)
+        14*() + 5*(2,3) + 2*(1,2) + 10*(1,2,3) + 13*(1,3,2) + 5*(1,3)
 
     :func:`GroupAlgebra` is just a short hand for a more general
     construction that covers, e.g., monoid algebras, additive group

src/sage/categories/algebra_functor.py

@@ -17,15 +17,15 @@
     Algebra of Dihedral group of order 6 as a permutation group
             over Rational Field
     sage: a = A.an_element(); a
-    () + (1,2) + 3*(1,2,3) + 2*(1,3)
+    () + (1,2) + 3*(1,2,3) + 2*(1,3,2)
 
 This space is endowed with an algebra structure, obtained by extending
 by bilinearity the multiplication of `G` to a multiplication on `RG`::
 
     sage: A in Algebras
     True
     sage: a * a
-    6*() + 9*(2,3) + 8*(1,2) + 8*(1,2,3) + 11*(1,3,2) + 7*(1,3)
+    14*() + 5*(2,3) + 2*(1,2) + 10*(1,2,3) + 13*(1,3,2) + 5*(1,3)
 
 In particular, the product of two basis elements is induced by the
 product of the corresponding elements of the group, and the unit of
@@ -230,12 +230,12 @@
     sage: a = A.an_element(); a
     () + 2*(3,4) + 3*(1,2) + (1,2)(3,4)
     sage: b = B.an_element(); b
-    () + (2,3,4) + 2*(1,2) + 3*(1,2,3,4)
+    () + (2,3,4) + 3*(1,2,4) + 2*(1,4)
     sage: B(a)
     () + 2*(3,4) + 3*(1,2) + (1,2)(3,4)
     sage: a * b  # a is automatically converted to an element of B
-    7*() + 4*(3,4) + 2*(2,3) + (2,3,4) + 5*(1,2) + 5*(1,2)(3,4) + 6*(1,2,3)
-     + 3*(1,2,3,4) + (1,3,2) + 3*(1,3,4,2) + 3*(1,3) + 9*(1,3,4)
+    () + 2*(3,4) + 2*(2,3) + (2,3,4) + 3*(1,2) + (1,2)(3,4) + 8*(1,2,4,3)
+     + 9*(1,2,4) + (1,3,2) + 3*(1,3,4,2) + 7*(1,4,3) + 11*(1,4)
     sage: parent(a * b)
     Symmetric group algebra of order 4 over Rational Field
 
@@ -244,7 +244,7 @@
     sage: A(b)
     Traceback (most recent call last):
     ...
-    TypeError: do not know how to make x (= () + (2,3,4) + 2*(1,2) + 3*(1,2,3,4))
+    TypeError: do not know how to make x (= () + (2,3,4) + 3*(1,2,4) + 2*(1,4))
      an element of self
      (=Algebra of Dihedral group of order 4 as a permutation group over Integer Ring)
 
@@ -339,11 +339,11 @@
     sage: kD4 in HopfAlgebras
     True
     sage: a = kD4.an_element(); a
-    () + 3*(1,2,3,4) + (1,3) + 2*(1,4)(2,3)
+    () + (1,3) + 2*(1,3)(2,4) + 3*(1,4,3,2)
     sage: a.antipode()
-    () + (1,3) + 3*(1,4,3,2) + 2*(1,4)(2,3)
+    () + 3*(1,2,3,4) + (1,3) + 2*(1,3)(2,4)
     sage: a.coproduct()
-    () # () + 3*(1,2,3,4) # (1,2,3,4) + (1,3) # (1,3) + 2*(1,4)(2,3) # (1,4)(2,3)
+    () # () + (1,3) # (1,3) + 2*(1,3)(2,4) # (1,3)(2,4) + 3*(1,4,3,2) # (1,4,3,2)
 
 Coercions from the base ring::
 
@@ -422,10 +422,10 @@
       From: Symmetric group algebra of order 3 over Integer Ring
       To: Symmetric group algebra of order 3 over Finite Field of size 5
     sage: a = 2 * A.an_element(); a
-    2*() + 2*(2,3) + 4*(1,2) + 6*(1,2,3)
+    2*() + 2*(2,3) + 6*(1,2,3) + 4*(1,3)
 
     sage: hh(a)
-    2*() + 2*(2,3) + 4*(1,2) + (1,2,3)
+    2*() + 2*(2,3) + (1,2,3) + 4*(1,3)
 
 Conversion from a formal sum::
 
@@ -626,9 +626,9 @@ def _apply_functor_to_morphism(self, f):
               To:   Symmetric group algebra of order 3 over Finite Field of size 5
 
             sage: a = 2 * A.an_element(); a
-            2*() + 2*(2,3) + 4*(1,2) + 6*(1,2,3)
+            2*() + 2*(2,3) + 6*(1,2,3) + 4*(1,3)
             sage: hh(a)
-            2*() + 2*(2,3) + 4*(1,2) + (1,2,3)
+            2*() + 2*(2,3) + (1,2,3) + 4*(1,3)
         """
         from sage.categories.rings import Rings
         domain   = self(f.domain())

src/sage/categories/finite_dimensional_lie_algebras_with_basis.py

@@ -306,7 +306,7 @@ def killing_form_matrix(self):
 
         @cached_method
         def structure_coefficients(self, include_zeros=False):
-            """
+            r"""
             Return the structure coefficients of ``self``.
 
             INPUT:
@@ -334,8 +334,16 @@ def structure_coefficients(self, include_zeros=False):
                 sage: S = GroupAlgebra(G, QQ)
                 sage: L = LieAlgebra(associative=S)
                 sage: L.structure_coefficients()
-                Finite family {((1,3), (2,3)): (1,2,3) - (1,3,2), ((2,3), (1,3,2)): -(1,2) + (1,3), ((1,3), (1,3,2)): -(2,3) + (1,2), ((1,3,2), (1,2)): -(2,3) + (1,3), ((1,3), (1,2,3)): (2,3) - (1,2), ((1,2,3), (1,2)): (2,3) - (1,3), ((1,3), (1,2)): -(1,2,3) + (1,3,2), ((2,3), (1,2)): (1,2,3) - (1,3,2), ((1,2,3), (2,3)): -(1,2) + (1,3)}
-                """
+                Finite family {((1,3), (2,3)): (1,2,3) - (1,3,2),
+                               ((2,3), (1,3,2)): -(1,2) + (1,3),
+                               ((1,3), (1,3,2)): -(2,3) + (1,2),
+                               ((1,3,2), (1,2)): -(2,3) + (1,3),
+                               ((1,3), (1,2,3)): (2,3) - (1,2),
+                               ((1,2,3), (1,2)): (2,3) - (1,3),
+                               ((1,3), (1,2)): -(1,2,3) + (1,3,2),
+                               ((2,3), (1,2)): (1,2,3) - (1,3,2),
+                               ((1,2,3), (2,3)): -(1,2) + (1,3)}
+            """
             d = {}
             B = self.basis()
             K = list(B.keys())

src/sage/categories/modules_with_basis.py

@@ -2155,7 +2155,7 @@ def _an_element_(self):
                     sage: A.an_element()
                     B[word: ] + 2*B[word: a] + 3*B[word: b] + B[word: bab]
                     sage: B.an_element()
-                    B[()] + B[(1,2)] + 3*B[(1,2,3)] + 2*B[(1,3)]
+                    B[()] + B[(1,2)] + 3*B[(1,2,3)] + 2*B[(1,3,2)]
                     sage: cartesian_product((A, B, A)).an_element()           # indirect doctest
                     2*B[(0, word: )] + 2*B[(0, word: a)] + 3*B[(0, word: b)]
                 """

src/sage/categories/sets_cat.py

@@ -1559,15 +1559,15 @@ def algebra(self, base_ring, category=None, **kwds):
                 sage: A.category()
                 Category of finite group algebras over Rational Field
                 sage: a = A.an_element(); a
-                () + 3*(1,2,3,4) + (1,3) + 2*(1,4)(2,3)
+                () + (1,3) + 2*(1,3)(2,4) + 3*(1,4,3,2)
 
             This space is endowed with an algebra structure, obtained
             by extending by bilinearity the multiplication of `G` to a
             multiplication on `RG`::
 
                 sage: a * a
-                6*() + 6*(2,4) + 3*(1,2)(3,4) + 8*(1,2,3,4) + 8*(1,3)
-                 + 9*(1,3)(2,4) + 2*(1,4,3,2) + 7*(1,4)(2,3)
+                6*() + 4*(2,4) + 3*(1,2)(3,4) + 12*(1,2,3,4) + 2*(1,3)
+                 + 13*(1,3)(2,4) + 6*(1,4,3,2) + 3*(1,4)(2,3)
 
             If `S` is a :class:`monoid <Monoids>`, the result is its
             monoid algebra `KS`::

src/sage/combinat/symmetric_group_algebra.py

@@ -93,7 +93,7 @@ def SymmetricGroupAlgebra(R, W, category=None):
         sage: SGA.group()
         Symmetric group of order 4! as a permutation group
         sage: SGA.an_element()
-        () + (2,3,4) + 2*(1,2) + 3*(1,2,3,4)
+        () + (2,3,4) + 3*(1,2,4) + 2*(1,4)
 
         sage: SGA = SymmetricGroupAlgebra(QQ, WeylGroup(["A",3], prefix='s')); SGA
         Symmetric group algebra of order 4 over Rational Field
@@ -247,7 +247,7 @@ def __init__(self, R, W, category):
             sage: G = SymmetricGroup(4).algebra(QQ)
             sage: S = SymmetricGroupAlgebra(QQ,4)
             sage: S(G.an_element())
-            [1, 2, 3, 4] + [1, 3, 4, 2] + 2*[2, 1, 3, 4] + 3*[2, 3, 4, 1]
+            [1, 2, 3, 4] + [1, 3, 4, 2] + 3*[2, 4, 3, 1] + 2*[4, 2, 3, 1]
             sage: G(S.an_element())
             () + 2*(3,4) + 3*(2,3) + (1,4,3,2)
 
@@ -936,7 +936,7 @@ def retract_okounkov_vershik(self, f, m):
 
             sage: G = SymmetricGroup(4).algebra(QQ)
             sage: G.retract_okounkov_vershik(G.an_element(), 3)
-            () + (2,3) + 2*(1,2) + 3*(1,2,3)
+            3*() + (2,3) + 3*(1,2)
 
         .. SEEALSO::
 

src/sage/groups/perm_gps/permgroup.py

@@ -802,10 +802,10 @@ def list(self):
 
             sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
             sage: G.list()
-            [(), (1,2), (1,2,3,4), (1,3)(2,4), (1,3,4), (2,3,4), (1,4,3,2),
-             (1,3,2,4), (1,3,4,2), (1,2,4,3), (1,4,2,3), (2,4,3), (1,4,3),
-             (1,4)(2,3), (1,4,2), (1,3,2), (1,3), (3,4), (2,4), (1,4), (2,3),
-             (1,2)(3,4), (1,2,3), (1,2,4)]
+            [(), (1,4)(2,3), (1,2)(3,4), (1,3)(2,4), (2,4,3), (1,4,2),
+             (1,2,3), (1,3,4), (2,3,4), (1,4,3), (1,2,4), (1,3,2), (3,4),
+             (1,4,2,3), (1,2), (1,3,2,4), (2,4), (1,4,3,2), (1,2,3,4),
+             (1,3), (2,3), (1,4), (1,2,4,3), (1,3,4,2)]
 
             sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b')); G
             Permutation Group with generators [('a','b')]
@@ -1597,7 +1597,7 @@ def base(self, seed=None):
         return [self._domain_from_gap[x] for x in self._gap_().StabChain(seed).BaseStabChain().sage()]
 
     def strong_generating_system(self, base_of_group=None, implementation="sage"):
-        """
+        r"""
         Return a Strong Generating System of ``self`` according the given
         base for the right action of ``self`` on itself.
 

src/sage/groups/perm_gps/permgroup_element.pyx

@@ -1,4 +1,4 @@
-"""
+r"""
 Permutation group elements
 
 AUTHORS:
@@ -11,27 +11,25 @@ AUTHORS:
 
 There are several ways to define a permutation group element:
 
--  Define a permutation group `G`, then use
-   ``G.gens()`` and multiplication \* to construct
-   elements.
+-  Define a permutation group `G`, then use ``G.gens()``
+   and multiplication ``*`` to construct elements.
 
--  Define a permutation group `G`, then use e.g.,
+-  Define a permutation group `G`, then use, e.g.,
    ``G([(1,2),(3,4,5)])`` to construct an element of the
    group. You could also use ``G('(1,2)(3,4,5)')``
 
--  Use e.g.,
+-  Use, e.g.,
    ``PermutationGroupElement([(1,2),(3,4,5)])`` or
    ``PermutationGroupElement('(1,2)(3,4,5)')`` to make a
    permutation group element with parent `S_5`.
 
+EXAMPLES:
 
-EXAMPLES: We illustrate construction of permutation using several
+We illustrate construction of permutation using several
 different methods.
 
 First we construct elements by multiplying together generators for
-a group.
-
-::
+a group::
 
     sage: G = PermutationGroup(['(1,2)(3,4)', '(3,4,5,6)'], canonicalize=False)
     sage: s = G.gens()
@@ -45,9 +43,7 @@ a group.
     Permutation Group with generators [(1,2)(3,4), (3,4,5,6)]
 
 Next we illustrate creation of a permutation using coercion into an
-already-created group.
-
-::
+already-created group::
 
     sage: g = G([(1,2),(3,5,6)])
     sage: g
@@ -58,9 +54,7 @@ already-created group.
     True
 
 We can also use a string or one-line notation to specify the
-permutation.
-
-::
+permutation::
 
     sage: h = G('(1,2)(3,5,6)')
     sage: i = G([2,1,5,4,6,3])
@@ -69,23 +63,24 @@ permutation.
 
 The Rubik's cube group::
 
-    sage: f= [(17,19,24,22),(18,21,23,20),(6,25,43,16),(7,28,42,13),(8,30,41,11)]
-    sage: b=[(33,35,40,38),(34,37,39,36),( 3, 9,46,32),( 2,12,47,29),( 1,14,48,27)]
-    sage: l=[( 9,11,16,14),(10,13,15,12),( 1,17,41,40),( 4,20,44,37),( 6,22,46,35)]
-    sage: r=[(25,27,32,30),(26,29,31,28),( 3,38,43,19),( 5,36,45,21),( 8,33,48,24)]
-    sage: u=[( 1, 3, 8, 6),( 2, 5, 7, 4),( 9,33,25,17),(10,34,26,18),(11,35,27,19)]
-    sage: d=[(41,43,48,46),(42,45,47,44),(14,22,30,38),(15,23,31,39),(16,24,32,40)]
-    sage: cube = PermutationGroup([f,b,l,r,u,d])
-    sage: F=cube.gens()[0]
-    sage: B=cube.gens()[1]
-    sage: L=cube.gens()[2]
-    sage: R=cube.gens()[3]
-    sage: U=cube.gens()[4]
-    sage: D=cube.gens()[5]
+    sage: f = [(17,19,24,22),(18,21,23,20),( 6,25,43,16),( 7,28,42,13),( 8,30,41,11)]
+    sage: b = [(33,35,40,38),(34,37,39,36),( 3, 9,46,32),( 2,12,47,29),( 1,14,48,27)]
+    sage: l = [( 9,11,16,14),(10,13,15,12),( 1,17,41,40),( 4,20,44,37),( 6,22,46,35)]
+    sage: r = [(25,27,32,30),(26,29,31,28),( 3,38,43,19),( 5,36,45,21),( 8,33,48,24)]
+    sage: u = [( 1, 3, 8, 6),( 2, 5, 7, 4),( 9,33,25,17),(10,34,26,18),(11,35,27,19)]
+    sage: d = [(41,43,48,46),(42,45,47,44),(14,22,30,38),(15,23,31,39),(16,24,32,40)]
+    sage: cube = PermutationGroup([f, b, l, r, u, d])
+    sage: F, B, L, R, U, D = cube.gens()
     sage: cube.order()
     43252003274489856000
     sage: F.order()
     4
+
+We create element of a permutation group of large degree::
+
+    sage: G = SymmetricGroup(30)
+    sage: s = G(srange(30,0,-1)); s
+    (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)
 """
 
 #*****************************************************************************

src/sage/modules/with_basis/representation.py

@@ -264,9 +264,9 @@ def _element_constructor_(self, x):
             sage: A = G.algebra(ZZ)
             sage: R = A.regular_representation()
             sage: x = A.an_element(); x
-            () + 3*(1,2,3,4) + (1,3) + 2*(1,4)(2,3)
+            () + (1,3) + 2*(1,3)(2,4) + 3*(1,4,3,2)
             sage: R(x)
-            () + 3*(1,2,3,4) + (1,3) + 2*(1,4)(2,3)
+            () + (1,3) + 2*(1,3)(2,4) + 3*(1,4,3,2)
         """
         if isinstance(x, Element) and x.parent() is self._module:
             return self._from_dict(x.monomial_coefficients(copy=False), remove_zeros=False)