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_orbits():
a = Permutation([2, 0, 1])
b = Permutation([2, 1, 0])
g = PermutationGroup([a, b])
assert g.orbit(0) == {0, 1, 2}
assert g.orbits() == [{0, 1, 2}]
assert g.is_transitive() and g.is_transitive(strict=False)
assert g.orbit_transversal(0) == \
[Permutation(
[0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
assert g.orbit_transversal(0, True) == \
[(0, Permutation([0, 1, 2])), (2, Permutation([2, 0, 1])),
(1, Permutation([1, 2, 0]))]
a = Permutation(list(range(1, 100)) + [0])
G = PermutationGroup([a])
assert [min(o) for o in G.orbits()] == [0]
G = PermutationGroup(rubik_cube_generators())
assert [min(o) for o in G.orbits()] == [0, 1]
assert not G.is_transitive() and not G.is_transitive(strict=False)
G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
assert not G.is_transitive() and G.is_transitive(strict=False)
assert PermutationGroup(
Permutation(3)).is_transitive(strict=False) is False