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_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_derived_subgroup():
a = Permutation([1, 0, 2, 4, 3])
b = Permutation([0, 1, 3, 2, 4])
G = PermutationGroup([a, b])
C = G.derived_subgroup()
assert C.order() == 3
assert C.is_normal(G)
assert C.is_subgroup(G, 0)
assert not G.is_subgroup(C, 0)
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)
C = G.derived_subgroup()
assert C.order() == 12
def test_eq():
a = [[1, 2, 0, 3, 4, 5], [1, 0, 2, 3, 4, 5], [2, 1, 0, 3, 4, 5], [
1, 2, 0, 3, 4, 5]]
a = [Permutation(p) for p in a + [[1, 2, 3, 4, 5, 0]]]
g = Permutation([1, 2, 3, 4, 5, 0])
G1, G2, G3 = [PermutationGroup(x) for x in [a[:2], a[2:4], [g, g**2]]]
assert G1.order() == G2.order() == G3.order() == 6
assert G1.is_subgroup(G2)
assert not G1.is_subgroup(G3)
G4 = PermutationGroup([Permutation([0, 1])])
assert not G1.is_subgroup(G4)
assert G4.is_subgroup(G1, 0)
assert PermutationGroup(g, g).is_subgroup(PermutationGroup(g))
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(4), 0)
assert SymmetricGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert not CyclicGroup(5).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)
assert CyclicGroup(3).is_subgroup(SymmetricGroup(3)*CyclicGroup(5), 0)