diff --git a/src/doc/en/thematic_tutorials/group_theory.rst b/src/doc/en/thematic_tutorials/group_theory.rst index d9ad8e9f00f..b5d5f38f718 100644 --- a/src/doc/en/thematic_tutorials/group_theory.rst +++ b/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)) diff --git a/src/sage/combinat/designs/incidence_structures.py b/src/sage/combinat/designs/incidence_structures.py index c76e6bf7aff..b5a747c58fa 100644 --- a/src/sage/combinat/designs/incidence_structures.py +++ b/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 [...] diff --git a/src/sage/graphs/bliss.pyx b/src/sage/graphs/bliss.pyx index dbe32840633..e44fda30b3f 100644 --- a/src/sage/graphs/bliss.pyx +++ b/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]) diff --git a/src/sage/graphs/generic_graph.py b/src/sage/graphs/generic_graph.py index 917019a392e..135adb80c3e 100644 --- a/src/sage/graphs/generic_graph.py +++ b/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:: diff --git a/src/sage/groups/finitely_presented_named.py b/src/sage/groups/finitely_presented_named.py index 76df3fad7dd..3a775237799 100644 --- a/src/sage/groups/finitely_presented_named.py +++ b/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: diff --git a/src/sage/groups/libgap_wrapper.pyx b/src/sage/groups/libgap_wrapper.pyx index 771280fdbf9..8f9207e7a04 100644 --- a/src/sage/groups/libgap_wrapper.pyx +++ b/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] diff --git a/src/sage/groups/perm_gps/permgroup.py b/src/sage/groups/perm_gps/permgroup.py index 616e138becf..82f18cd2755 100644 --- a/src/sage/groups/perm_gps/permgroup.py +++ b/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 diff --git a/src/sage/groups/perm_gps/permgroup_morphism.py b/src/sage/groups/perm_gps/permgroup_morphism.py index 69f12016e60..b00f6ce5c69 100644 --- a/src/sage/groups/perm_gps/permgroup_morphism.py +++ b/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:: diff --git a/src/sage/libs/gap/assigned_names.py b/src/sage/libs/gap/assigned_names.py index 216bee639ec..d7429ba007f 100644 --- a/src/sage/libs/gap/assigned_names.py +++ b/src/sage/libs/gap/assigned_names.py @@ -124,7 +124,7 @@ def list_functions(): sage: 'SubdirectProduct' in FUNCTIONS # indirect doctest True """ - fnames = set(GLOBALS).difference(KEYWORDS) + fnames = set(GLOBALS) - {*KEYWORDS, 'AtlasOfGroupRepresentationsInfo'} # accessing the AtlasOfGroupRepresentationsInfo global variable requires internet access and will timeout if the server is down documented = Filtered(list(fnames), IsDocumentedWord) return tuple(sorted(documented.sage())) diff --git a/src/sage/topology/simplicial_complex.py b/src/sage/topology/simplicial_complex.py index 3cd267d135a..165b06d381b 100644 --- a/src/sage/topology/simplicial_complex.py +++ b/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']]) diff --git a/src/sage/topology/simplicial_complex_examples.py b/src/sage/topology/simplicial_complex_examples.py index 55a6b943902..e54a0586977 100644 --- a/src/sage/topology/simplicial_complex_examples.py +++ b/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