Add internet doctest marker to various places

Author: user202729

src/doc/en/thematic_tutorials/group_theory.rst

@@ -978,6 +978,7 @@ to do this.  The symmetric group on 7 symbols, `S_7`, has order
 `7! = 5040` and is divisible by `2^4 = 16`.  Let's find one example
 of a subgroup of permutations on 4 symbols with order 16::
 
+    sage: # optional - internet
     sage: G = SymmetricGroup(7)
     sage: subgroups = G.conjugacy_classes_subgroups()
     sage: list(map(order, subgroups))

src/sage/combinat/designs/incidence_structures.py

@@ -1865,7 +1865,7 @@ def automorphism_group(self):
             sage: P = designs.DesarguesianProjectivePlaneDesign(2); P
             (7,3,1)-Balanced Incomplete Block Design
             sage: G = P.automorphism_group()
-            sage: G.is_isomorphic(PGL(3,2))
+            sage: G.is_isomorphic(PGL(3,2))  # optional - internet
             True
             sage: G
             Permutation Group with generators [...]

src/sage/graphs/bliss.pyx

@@ -752,7 +752,7 @@ cpdef automorphism_group(G, partition=None, use_edge_labels=True) noexcept:
         sage: G = graphs.HeawoodGraph()
         sage: p = G.bipartite_sets()
         sage: A = G.automorphism_group(partition=[list(p[0]), list(p[1])])
-        sage: automorphism_group(G, partition=p).is_isomorphic(A)
+        sage: automorphism_group(G, partition=p).is_isomorphic(A)  # optional - internet
         True
 
         sage: G = graphs.CompleteMultipartiteGraph([5, 7, 11])

src/sage/graphs/generic_graph.py

@@ -24700,7 +24700,7 @@ def automorphism_group(self, partition=None, verbosity=0,
             sage: A5 = AlternatingGroup(5)
             sage: Z2 = CyclicPermutationGroup(2)
             sage: H = A5.direct_product(Z2)[0] #see documentation for direct_product to explain the [0]
-            sage: G.is_isomorphic(H)
+            sage: G.is_isomorphic(H)  # optional - internet
             True
 
         Multigraphs::

src/sage/groups/finitely_presented_named.py

@@ -480,7 +480,7 @@ def AlternatingPresentation(n) -> FinitelyPresentedGroup:
     EXAMPLES::
 
         sage: A6 = groups.presentation.Alternating(6)
-        sage: A6.as_permutation_group().is_isomorphic(AlternatingGroup(6)), A6.order()
+        sage: A6.as_permutation_group().is_isomorphic(AlternatingGroup(6)), A6.order()  # optional - internet
         (True, 360)
 
     TESTS:

src/sage/groups/libgap_wrapper.pyx

@@ -334,7 +334,7 @@ class ParentLibGAP(SageObject):
 
         EXAMPLES::
 
-            sage: SL(2,GF(49)).minimal_normal_subgroups()
+            sage: SL(2,GF(49)).minimal_normal_subgroups()  # optional - internet
             [Subgroup with 1 generators (
              [6 0]
              [0 6]
@@ -352,7 +352,7 @@ class ParentLibGAP(SageObject):
 
         EXAMPLES::
 
-            sage: SL(2,GF(49)).minimal_normal_subgroups()
+            sage: SL(2,GF(49)).minimal_normal_subgroups()  # optional - internet
             [Subgroup with 1 generators (
              [6 0]
              [0 6]

src/sage/groups/perm_gps/permgroup.py

@@ -1582,7 +1582,7 @@ def disjoint_direct_product_decomposition(self):
 
         Counting the number of "connected" permutation groups of degree `n`::
 
-            sage: seq = [sum(1 for G in SymmetricGroup(n).conjugacy_classes_subgroups() if len(G.disjoint_direct_product_decomposition()) == 1) for n in range(1,8)]; seq
+            sage: seq = [sum(1 for G in SymmetricGroup(n).conjugacy_classes_subgroups() if len(G.disjoint_direct_product_decomposition()) == 1) for n in range(1,8)]; seq  # optional - internet
             [1, 1, 2, 6, 6, 27, 20]
             sage: oeis(seq) # optional -- internet
             0: A005226: Number of atomic species of degree n; also number of connected permutation groups of degree n.
@@ -3162,7 +3162,7 @@ def as_finitely_presented_group(self, reduced=False):
             sage: A = AlternatingGroup(5).as_finitely_presented_group().gap()
             sage: ctab = A.CosetTable(A.Subgroup([]))
             sage: gen_ls = gap.List(ctab, gap.PermList)
-            sage: PermutationGroup(gen_ls).is_isomorphic(AlternatingGroup(5))
+            sage: PermutationGroup(gen_ls).is_isomorphic(AlternatingGroup(5))  # optional - internet
             True
 
         AUTHORS:
@@ -3252,7 +3252,7 @@ def commutator(self, other=None):
             sage: G = SymmetricGroup(5)
             sage: H = CyclicPermutationGroup(5)
             sage: C = G.commutator(H)
-            sage: C.is_isomorphic(AlternatingGroup(5))
+            sage: C.is_isomorphic(AlternatingGroup(5))  # optional - internet
             True
 
         An abelian group will have a trivial commutator.  ::
@@ -4925,6 +4925,8 @@ def normal_subgroups(self):
             sage: G = PSL(2,7)
             sage: D = G.direct_product(G)
             sage: H = D[0]
+
+            sage: # optional - internet
             sage: NH = H.normal_subgroups()
             sage: len(NH)
             4

src/sage/groups/perm_gps/permgroup_morphism.py

@@ -81,7 +81,7 @@ def kernel(self):
             sage: D = G.direct_product(G)
             sage: H = D[0]
             sage: pr1 = D[3]
-            sage: G.is_isomorphic(pr1.kernel())
+            sage: G.is_isomorphic(pr1.kernel())  # optional - internet
             True
         """
         return self.domain().subgroup(gap_group=self._libgap_().Kernel())
@@ -113,7 +113,7 @@ def image(self, J):
             sage: pr1.image(G)
             Subgroup generated by [(3,7,5)(4,8,6), (1,2,6)(3,4,8)] of
              (The projective special linear group of degree 2 over Finite Field of size 7)
-            sage: G.is_isomorphic(pr1.image(G))
+            sage: G.is_isomorphic(pr1.image(G))  # optional - internet
             True
 
         Check that :issue:`28324` is fixed::

src/sage/topology/simplicial_complex.py

@@ -4296,7 +4296,7 @@ def automorphism_group(self):
             True
 
             sage: P = simplicial_complexes.RealProjectivePlane()
-            sage: P.automorphism_group().is_isomorphic(AlternatingGroup(5))             # needs sage.groups
+            sage: P.automorphism_group().is_isomorphic(AlternatingGroup(5))             # needs sage.groups  # optional - internet
             True
 
             sage: Z = SimplicialComplex([['1','2'],['2','3','a']])

src/sage/topology/simplicial_complex_examples.py

@@ -583,7 +583,7 @@ def QuaternionicProjectivePlane():
 
     Checking its automorphism group::
 
-        sage: HP2.automorphism_group().is_isomorphic(AlternatingGroup(5))               # needs sage.groups
+        sage: HP2.automorphism_group().is_isomorphic(AlternatingGroup(5))               # needs sage.groups  # optional - internet
         True
     """
     from sage.groups.perm_gps.permgroup import PermutationGroup