def test_is_normal():
gens_s5 = [Permutation(p) for p in [[1, 2, 3, 4, 0], [2, 1, 4, 0, 3]]]
G1 = PermutationGroup(gens_s5)
assert G1.order() == 120
gens_a5 = [Permutation(p) for p in [[1, 0, 3, 2, 4], [2, 1, 4, 3, 0]]]
G2 = PermutationGroup(gens_a5)
assert G2.order() == 60
assert G2.is_normal(G1)
gens3 = [Permutation(p) for p in [[2, 1, 3, 0, 4], [1, 2, 0, 3, 4]]]
G3 = PermutationGroup(gens3)
assert not G3.is_normal(G1)
assert G3.order() == 12
G4 = G1.normal_closure(G3.generators)
assert G4.order() == 60
gens5 = [Permutation(p) for p in [[1, 2, 3, 0, 4], [1, 2, 0, 3, 4]]]
G5 = PermutationGroup(gens5)
assert G5.order() == 24
G6 = G1.normal_closure(G5.generators)
assert G6.order() == 120
assert G1.is_subgroup(G6)
assert not G1.is_subgroup(G4)
assert G2.is_subgroup(G4)
s4 = PermutationGroup(Permutation(0, 1, 2, 3), Permutation(3)(0, 1))
s6 = PermutationGroup(Permutation(0, 1, 2, 3, 5), Permutation(5)(0, 1))
assert s6.is_normal(s4, strict=False)
assert not s4.is_normal(s6, strict=False)
def test_PermutationGroup():
assert PermutationGroup() == PermutationGroup(Permutation())
a = Permutation(1, 2)
b = Permutation(2, 3, 1)
G = PermutationGroup(a, b, degree=5)
assert G.contains(G[0])
A = AlternatingGroup(4)
A.schreier_sims()
assert A.base == [0, 1]
assert A.basic_stabilizers == [
PermutationGroup(Permutation(0, 1, 2), Permutation(1, 2, 3)),
PermutationGroup(Permutation(1, 2, 3))
]
D = DihedralGroup(12)
assert D.is_primitive(randomized=False) is False
D = DihedralGroup(10)
assert D.is_primitive() is False
p = Permutation(0, 1, 2, 3, 4, 5)
G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)])
G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)])
G3 = PermutationGroup([p, p**2])
assert G1.order() == G2.order() == G3.order() == 6
assert G1.is_subgroup(G2) is True
assert G1.is_subgroup(G3) is False
a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
G = PermutationGroup([a, b])
assert G.make_perm([0, 1, 0]) == Permutation(0, 2, 3, 1)
S = SymmetricGroup(5)
base, strong_gens = S.schreier_sims_random()
assert _verify_bsgs(S, base, strong_gens)
D = DihedralGroup(4)
assert D.strong_gens == [
Permutation(0, 1, 2, 3),
Permutation(0, 3)(1, 2),
Permutation(1, 3)
]
a = Permutation([1, 2, 0])
b = Permutation([1, 0, 2])
G = PermutationGroup([a, b])
assert G.transitivity_degree == 3
a = Permutation([1, 2, 0, 4, 5, 6, 3])
G = PermutationGroup([a])
assert G.orbit(0) == {0, 1, 2}
assert G.orbit([0, 4], 'union') == {0, 1, 2, 3, 4, 5, 6}
assert G.orbit([0, 4], 'sets') == {(0, 3), (0, 4), (0, 5), (0, 6), (1, 3),
(1, 4), (1, 5), (1, 6), (2, 3), (2, 4),
(2, 5), (2, 6)}
assert G.orbit([0, 4], 'tuples') == {(0, 3), (0, 4), (0, 5), (0, 6),
(1, 3), (1, 4), (1, 5), (1, 6),
(2, 3), (2, 4), (2, 5), (2, 6)}
def test_rubik():
G = PermutationGroup(rubik_cube_generators())
assert G.order() == 43252003274489856000
G1 = PermutationGroup(G[:3])
assert G1.order() == 170659735142400
assert not G1.is_normal(G)
G2 = G.normal_closure(G1.generators)
assert G2.is_subgroup(G)
def test_stabilizer():
S = SymmetricGroup(2)
H = S.stabilizer(0)
assert H.generators == [Permutation(1)]
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
G = PermutationGroup([a, b])
G0 = G.stabilizer(0)
assert G0.order() == 60
gens_cube = [[1, 3, 5, 7, 0, 2, 4, 6], [1, 3, 0, 2, 5, 7, 4, 6]]
gens = [Permutation(p) for p in gens_cube]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 6
G2_1 = G2.stabilizer(1)
v = list(G2_1.generate(af=True))
assert v == [[0, 1, 2, 3, 4, 5, 6, 7], [3, 1, 2, 0, 7, 5, 6, 4]]
gens = (
(1, 2, 0, 4, 5, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19),
(0, 1, 2, 3, 4, 5, 19, 6, 8, 9, 10, 11, 12, 13, 14,
15, 16, 7, 17, 18),
(0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 16, 11, 12, 13, 14, 15, 8, 17, 10, 19))
gens = [Permutation(p) for p in gens]
G = PermutationGroup(gens)
G2 = G.stabilizer(2)
assert G2.order() == 181440
S = SymmetricGroup(3)
assert [G.order() for G in S.basic_stabilizers] == [6, 2]
def test_coset_factor():
a = Permutation([0, 2, 1])
G = PermutationGroup([a])
c = Permutation([2, 1, 0])
assert not G.coset_factor(c)
assert G.coset_rank(c) is None
a = Permutation([2, 0, 1, 3, 4, 5])
b = Permutation([2, 1, 3, 4, 5, 0])
g = PermutationGroup([a, b])
assert g.order() == 360
d = Permutation([1, 0, 2, 3, 4, 5])
assert not g.coset_factor(d.array_form)
assert not g.contains(d)
assert Permutation(2) in G
c = Permutation([1, 0, 2, 3, 5, 4])
v = g.coset_factor(c, True)
tr = g.basic_transversals
p = Permutation.rmul(*[tr[i][v[i]] for i in range(len(g.base))])
assert p == c
v = g.coset_factor(c)
p = Permutation.rmul(*v)
assert p == c
assert g.contains(c)
G = PermutationGroup([Permutation([2, 1, 0])])
p = Permutation([1, 0, 2])
assert G.coset_factor(p) == []
def test_rubik1():
gens = rubik_cube_generators()
gens1 = [gens[-1]] + [p**2 for p in gens[1:]]
G1 = PermutationGroup(gens1)
assert G1.order() == 19508428800
gens2 = [p**2 for p in gens]
G2 = PermutationGroup(gens2)
assert G2.order() == 663552
assert G2.is_subgroup(G1, 0)
C1 = G1.derived_subgroup()
assert C1.order() == 4877107200
assert C1.is_subgroup(G1, 0)
assert not G2.is_subgroup(C1, 0)
G = RubikGroup(2)
assert G.order() == 3674160
pytest.raises(ValueError, lambda: RubikGroup(0))
pytest.raises(ValueError, lambda: rubik(1))
G = RubikGroup(3)
assert G.order() == 43252003274489856000
def test_order():
a = Permutation([2, 0, 1, 3, 4, 5, 6, 7, 8, 9])
b = Permutation([2, 1, 3, 4, 5, 6, 7, 8, 9, 0])
g = PermutationGroup([a, b])
assert g.order() == 1814400
assert PermutationGroup().order() == 1