def test_verify_bsgs():
S = SymmetricGroup(5)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert _verify_bsgs(S, base, strong_gens) is True
assert _verify_bsgs(S, base[:-1], strong_gens) is False
assert _verify_bsgs(S, base, S.generators) is False
def test_verify_bsgs():
S = SymmetricGroup(5)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert _verify_bsgs(S, base, strong_gens) is True
assert _verify_bsgs(S, base[:-1], strong_gens) is False
assert _verify_bsgs(S, base, S.generators) is False
def test_baseswap():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert base == [0, 1, 2]
deterministic = S.baseswap(base, strong_gens, 1, randomized=False)
randomized = S.baseswap(base, strong_gens, 1)
assert deterministic[0] == [0, 2, 1]
assert _verify_bsgs(S, deterministic[0], deterministic[1]) is True
assert randomized[0] == [0, 2, 1]
assert _verify_bsgs(S, randomized[0], randomized[1]) is True
def test_baseswap():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
assert base == [0, 1, 2]
deterministic = S.baseswap(base, strong_gens, 1, randomized=False)
randomized = S.baseswap(base, strong_gens, 1)
assert deterministic[0] == [0, 2, 1]
assert _verify_bsgs(S, deterministic[0], deterministic[1]) is True
assert randomized[0] == [0, 2, 1]
assert _verify_bsgs(S, randomized[0], randomized[1]) is True
def test_remove_gens():
S = SymmetricGroup(10)
base, strong_gens = S.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(S, base, new_gens) is True
A = AlternatingGroup(7)
base, strong_gens = A.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(A, base, new_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(D, base, new_gens) is True
def test_schreier_sims_random():
assert sorted(tetra.pgroup.base) == [0, 1]
S = SymmetricGroup(3)
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([1, 0, 2]),
Permutation([0, 2, 1])]
assert S.schreier_sims_random(base, strong_gens, 5) == (base, strong_gens)
S = SymmetricGroup(5)
_random_prec = {'g': [Permutation(1, 4), Permutation(0, 3)(1, 4),
Permutation(0, 1, 2, 3, 4), Permutation(0, 1, 2)(3, 4),
Permutation(4)(0, 1)(2, 3), Permutation(0, 1, 4, 3),
Permutation(0, 1)(2, 4), Permutation(0, 3, 4, 2, 1),
Permutation(0, 4, 1)(2, 3)]}
base, strong_gens = S.schreier_sims_random(consec_succ=5,
_random_prec=_random_prec)
assert _verify_bsgs(S, base, strong_gens) is True
D = DihedralGroup(3)
_random_prec = {'g': [Permutation([2, 0, 1]), Permutation([1, 2, 0]),
Permutation([1, 0, 2])]}
base = [0, 1]
strong_gens = [Permutation([1, 2, 0]), Permutation([2, 1, 0]),
Permutation([0, 2, 1])]
assert D.schreier_sims_random([], D.generators, 2,
_random_prec=_random_prec) == (base, strong_gens)
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_remove_gens():
S = SymmetricGroup(10)
base, strong_gens = S.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(S, base, new_gens) is True
A = AlternatingGroup(7)
base, strong_gens = A.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(A, base, new_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(D, base, new_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental()
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
_, transversals = _orbits_transversals_from_bsgs(base, strong_gens_distr)
new_gens = _remove_gens(base, strong_gens, transversals, strong_gens_distr)
assert _verify_bsgs(D, base, new_gens) is True
def test_remove_gens():
S = SymmetricGroup(10)
base, strong_gens = S.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(S, base, new_gens) is True
A = AlternatingGroup(7)
base, strong_gens = A.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(A, base, new_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental()
new_gens = _remove_gens(base, strong_gens)
assert _verify_bsgs(D, base, new_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental()
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
_, transversals = _orbits_transversals_from_bsgs(base, strong_gens_distr)
new_gens = _remove_gens(base, strong_gens, transversals, strong_gens_distr)
assert _verify_bsgs(D, base, new_gens) is True
def test_schreier_sims_incremental():
identity = Permutation([0, 1, 2, 3, 4])
TrivialGroup = PermutationGroup([identity])
base, strong_gens = TrivialGroup.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(TrivialGroup, base, strong_gens) is True
S = SymmetricGroup(5)
base, strong_gens = S.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(S, base, strong_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental(base=[1])
assert _verify_bsgs(D, base, strong_gens) is True
A = AlternatingGroup(7)
gens = A.generators[:]
gen0 = gens[0]
gen1 = gens[1]
gen1 = rmul(gen1, ~gen0)
gen0 = rmul(gen0, gen1)
gen1 = rmul(gen0, gen1)
base, strong_gens = A.schreier_sims_incremental(base=[0, 1], gens=gens)
assert _verify_bsgs(A, base, strong_gens) is True
C = CyclicGroup(11)
gen = C.generators[0]
base, strong_gens = C.schreier_sims_incremental(gens=[gen**3])
assert _verify_bsgs(C, base, strong_gens) is True
def test_schreier_sims_incremental():
identity = Permutation([0, 1, 2, 3, 4])
TrivialGroup = PermutationGroup([identity])
base, strong_gens = TrivialGroup.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(TrivialGroup, base, strong_gens) is True
S = SymmetricGroup(5)
base, strong_gens = S.schreier_sims_incremental(base=[0, 1, 2])
assert _verify_bsgs(S, base, strong_gens) is True
D = DihedralGroup(2)
base, strong_gens = D.schreier_sims_incremental(base=[1])
assert _verify_bsgs(D, base, strong_gens) is True
A = AlternatingGroup(7)
gens = A.generators[:]
gen0 = gens[0]
gen1 = gens[1]
gen1 = rmul(gen1, ~gen0)
gen0 = rmul(gen0, gen1)
gen1 = rmul(gen0, gen1)
base, strong_gens = A.schreier_sims_incremental(base=[0, 1], gens=gens)
assert _verify_bsgs(A, base, strong_gens) is True
C = CyclicGroup(11)
gen = C.generators[0]
base, strong_gens = C.schreier_sims_incremental(gens=[gen**3])
assert _verify_bsgs(C, base, strong_gens) is True
def _subgroup_search(i, j, k):
def prop_true(x):
return True
def prop_fix_points(x):
return [x(point) for point in points] == points
def prop_comm_g(x):
return rmul(x, g) == rmul(g, x)
def prop_even(x):
return x.is_even
for i in range(i, j, k):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
Sym = S.subgroup_search(prop_true)
assert Sym.is_subgroup(S)
Alt = S.subgroup_search(prop_even)
assert Alt.is_subgroup(A)
Sym = S.subgroup_search(prop_true, init_subgroup=C)
assert Sym.is_subgroup(S)
points = [7]
assert S.stabilizer(7).is_subgroup(S.subgroup_search(prop_fix_points))
points = [3, 4]
assert S.stabilizer(3).stabilizer(4).is_subgroup(
S.subgroup_search(prop_fix_points))
points = [3, 5]
fix35 = A.subgroup_search(prop_fix_points)
points = [5]
fix5 = A.subgroup_search(prop_fix_points)
assert A.subgroup_search(prop_fix_points,
init_subgroup=fix35).is_subgroup(fix5)
base, strong_gens = A.schreier_sims_incremental()
g = A.generators[0]
comm_g = \
A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens)
assert _verify_bsgs(comm_g, base, comm_g.generators) is True
assert [prop_comm_g(gen) is True for gen in comm_g.generators]
def _subgroup_search(i, j, k):
def prop_true(x):
return True
def prop_fix_points(x):
return [x(point) for point in points] == points
def prop_comm_g(x):
return rmul(x, g) == rmul(g, x)
def prop_even(x):
return x.is_even
for i in range(i, j, k):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
Sym = S.subgroup_search(prop_true)
assert Sym.is_subgroup(S)
Alt = S.subgroup_search(prop_even)
assert Alt.is_subgroup(A)
Sym = S.subgroup_search(prop_true, init_subgroup=C)
assert Sym.is_subgroup(S)
points = [7]
assert S.stabilizer(7).is_subgroup(S.subgroup_search(prop_fix_points))
points = [3, 4]
assert S.stabilizer(3).stabilizer(4).is_subgroup(
S.subgroup_search(prop_fix_points))
points = [3, 5]
fix35 = A.subgroup_search(prop_fix_points)
points = [5]
fix5 = A.subgroup_search(prop_fix_points)
assert A.subgroup_search(prop_fix_points,
init_subgroup=fix35).is_subgroup(fix5)
base, strong_gens = A.schreier_sims_incremental()
g = A.generators[0]
comm_g = \
A.subgroup_search(prop_comm_g, base=base, strong_gens=strong_gens)
assert _verify_bsgs(comm_g, base, comm_g.generators) is True
assert [prop_comm_g(gen) is True for gen in comm_g.generators]
