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_commutator():
# the commutator of the trivial group and the trivial group is trivial
S = SymmetricGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert S.commutator(triv, triv).is_subgroup(triv)
# the commutator of the trivial group and any other group is again trivial
A = AlternatingGroup(3)
assert S.commutator(triv, A).is_subgroup(triv)
# the commutator is commutative
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
assert S.commutator(A, D).is_subgroup(S.commutator(D, A))
# the commutator of an abelian group is trivial
S = SymmetricGroup(7)
A1 = AbelianGroup(2, 5)
A2 = AbelianGroup(3, 4)
triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])])
assert S.commutator(A1, A1).is_subgroup(triv)
assert S.commutator(A2, A2).is_subgroup(triv)
# examples calculated by hand
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert S.commutator(A, S).is_subgroup(A)
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
S = SymmetricGroup(10)
N_eps = 10
_random_prec = {'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
_random_prec = {'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])}
assert A.is_alt_sym(_random_prec=_random_prec) is False
def test_is_alt_sym():
G = DihedralGroup(10)
assert G.is_alt_sym() is False
S = SymmetricGroup(10)
N_eps = 10
_random_prec = {
'N_eps': N_eps,
0: Permutation([[2], [1, 4], [0, 6, 7, 8, 9, 3, 5]]),
1: Permutation([[1, 8, 7, 6, 3, 5, 2, 9], [0, 4]]),
2: Permutation([[5, 8], [4, 7], [0, 1, 2, 3, 6, 9]]),
3: Permutation([[3], [0, 8, 2, 7, 4, 1, 6, 9, 5]]),
4: Permutation([[8], [4, 7, 9], [3, 6], [0, 5, 1, 2]]),
5: Permutation([[6], [0, 2, 4, 5, 1, 8, 3, 9, 7]]),
6: Permutation([[6, 9, 8], [4, 5], [1, 3, 7], [0, 2]]),
7: Permutation([[4], [0, 2, 9, 1, 3, 8, 6, 5, 7]]),
8: Permutation([[1, 5, 6, 3], [0, 2, 7, 8, 4, 9]]),
9: Permutation([[8], [6, 7], [2, 3, 4, 5], [0, 1, 9]])
}
assert S.is_alt_sym(_random_prec=_random_prec) is True
A = AlternatingGroup(10)
_random_prec = {
'N_eps': N_eps,
0: Permutation([[1, 6, 4, 2, 7, 8, 5, 9, 3], [0]]),
1: Permutation([[1], [0, 5, 8, 4, 9, 2, 3, 6, 7]]),
2: Permutation([[1, 9, 8, 3, 2, 5], [0, 6, 7, 4]]),
3: Permutation([[6, 8, 9], [4, 5], [1, 3, 7, 2], [0]]),
4: Permutation([[8], [5], [4], [2, 6, 9, 3], [1], [0, 7]]),
5: Permutation([[3, 6], [0, 8, 1, 7, 5, 9, 4, 2]]),
6: Permutation([[5], [2, 9], [1, 8, 3], [0, 4, 7, 6]]),
7: Permutation([[1, 8, 4, 7, 2, 3], [0, 6, 9, 5]]),
8: Permutation([[5, 8, 7], [3], [1, 4, 2, 6], [0, 9]]),
9: Permutation([[4, 9, 6], [3, 8], [1, 2], [0, 5, 7]])
}
assert A.is_alt_sym(_random_prec=_random_prec) is False
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)
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_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_normal_closure():
# the normal closure of the trivial group is trivial
S = SymmetricGroup(3)
identity = Permutation([0, 1, 2])
closure = S.normal_closure(identity)
assert closure.is_trivial
# the normal closure of the entire group is the entire group
A = AlternatingGroup(4)
assert A.normal_closure(A).is_subgroup(A)
# brute-force verifications for subgroups
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
C = CyclicGroup(i)
for gp in (A, D, C):
assert _verify_normal_closure(S, gp)
# brute-force verifications for all elements of a group
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_normal_closure(S, element)
# small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp2.is_subgroup(gp, 0) and gp2.degree == gp.degree:
assert _verify_normal_closure(gp, gp2)
def test_random_stab():
S = SymmetricGroup(5)
_random_el = Permutation([1, 3, 2, 0, 4])
_random_prec = {'rand': _random_el}
g = S.random_stab(2, _random_prec=_random_prec)
assert g == Permutation([1, 3, 2, 0, 4])
h = S.random_stab(1)
assert h(1) == 1
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_random_stab():
S = SymmetricGroup(5)
_random_el = Permutation([1, 3, 2, 0, 4])
_random_prec = {'rand': _random_el}
g = S.random_stab(2, _random_prec=_random_prec)
assert g == Permutation([1, 3, 2, 0, 4])
h = S.random_stab(1)
assert h(1) == 1
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_minimal_block():
D = DihedralGroup(6)
block_system = D.minimal_block([0, 3])
for i in range(3):
assert block_system[i] == block_system[i + 3]
S = SymmetricGroup(6)
assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0]
assert tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0]
def test_schreier_vector():
G = CyclicGroup(50)
v = [0]*50
v[23] = -1
assert G.schreier_vector(23) == v
H = DihedralGroup(8)
assert H.schreier_vector(2) == [0, 1, -1, 0, 0, 1, 0, 0]
L = SymmetricGroup(4)
assert L.schreier_vector(1) == [1, -1, 0, 0]
def test_minimal_block():
D = DihedralGroup(6)
block_system = D.minimal_block([0, 3])
for i in range(3):
assert block_system[i] == block_system[i + 3]
S = SymmetricGroup(6)
assert S.minimal_block([0, 1]) == [0, 0, 0, 0, 0, 0]
assert Tetra.pgroup.minimal_block([0, 1]) == [0, 0, 0, 0]
def test_schreier_vector():
G = CyclicGroup(50)
v = [0] * 50
v[23] = -1
assert G.schreier_vector(23) == v
H = DihedralGroup(8)
assert H.schreier_vector(2) == [0, 1, -1, 0, 0, 1, 0, 0]
L = SymmetricGroup(4)
assert L.schreier_vector(1) == [1, -1, 0, 0]
def test_verify_normal_closure():
# verified by GAP
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert _verify_normal_closure(S, A, closure=A)
S = SymmetricGroup(5)
A = AlternatingGroup(5)
C = CyclicGroup(5)
assert _verify_normal_closure(S, A, closure=A)
assert _verify_normal_closure(S, C, closure=A)
def test_normal_closure():
# the normal closure of the trivial group is trivial
S = SymmetricGroup(3)
identity = Permutation([0, 1, 2])
closure = S.normal_closure(identity)
assert closure.is_trivial
# the normal closure of the entire group is the entire group
A = AlternatingGroup(4)
assert A.normal_closure(A).is_subgroup(A)
# brute-force verifications for subgroups
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
C = CyclicGroup(i)
for gp in (A, D, C):
assert _verify_normal_closure(S, gp)
# brute-force verifications for all elements of a group
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_normal_closure(S, element)
# small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp2.is_subgroup(gp, 0) and gp2.degree == gp.degree:
assert _verify_normal_closure(gp, gp2)
def test_pointwise_stabilizer():
S = SymmetricGroup(2)
stab = S.pointwise_stabilizer([0])
assert stab.generators == [Permutation(1)]
S = SymmetricGroup(5)
points = []
stab = S
for point in (2, 0, 3, 4, 1):
stab = stab.stabilizer(point)
points.append(point)
assert S.pointwise_stabilizer(points).is_subgroup(stab)
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_SymmetricGroup():
G = SymmetricGroup(5)
elements = list(G.generate())
assert (G.generators[0]).size == 5
assert len(elements) == 120
assert G.is_solvable is False
assert G.is_abelian is False
assert G.is_nilpotent is False
assert G.is_transitive() is True
H = SymmetricGroup(1)
assert H.order() == 1
L = SymmetricGroup(2)
assert L.order() == 2
def test_derived_series():
# the derived series of the trivial group consists only of the trivial group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.derived_series()[0].is_subgroup(triv)
# the derived series for a simple group consists only of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.derived_series()[0].is_subgroup(A)
# the derived series for S_4 is S_4 > A_4 > K_4 > triv
S = SymmetricGroup(4)
series = S.derived_series()
assert series[1].is_subgroup(AlternatingGroup(4))
assert series[2].is_subgroup(DihedralGroup(2))
assert series[3].is_trivial
def test_lower_central_series():
# the lower central series of the trivial group consists of the trivial
# group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.lower_central_series()[0].is_subgroup(triv)
# the lower central series of a simple group consists of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.lower_central_series()[0].is_subgroup(A)
# GAP-verified example
S = SymmetricGroup(6)
series = S.lower_central_series()
assert len(series) == 2
assert series[1].is_subgroup(AlternatingGroup(6))
def test_lower_central_series():
# the lower central series of the trivial group consists of the trivial
# group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.lower_central_series()[0].is_subgroup(triv)
# the lower central series of a simple group consists of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.lower_central_series()[0].is_subgroup(A)
# GAP-verified example
S = SymmetricGroup(6)
series = S.lower_central_series()
assert len(series) == 2
assert series[1].is_subgroup(AlternatingGroup(6))
def test_derived_series():
# the derived series of the trivial group consists only of the trivial group
triv = PermutationGroup([Permutation([0, 1, 2])])
assert triv.derived_series()[0].is_subgroup(triv)
# the derived series for a simple group consists only of the group itself
for i in (5, 6, 7):
A = AlternatingGroup(i)
assert A.derived_series()[0].is_subgroup(A)
# the derived series for S_4 is S_4 > A_4 > K_4 > triv
S = SymmetricGroup(4)
series = S.derived_series()
assert series[1].is_subgroup(AlternatingGroup(4))
assert series[2].is_subgroup(DihedralGroup(2))
assert series[3].is_trivial
def test_verify_centralizer():
# verified by GAP
S = SymmetricGroup(3)
A = AlternatingGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert _verify_centralizer(S, S, centr=triv)
assert _verify_centralizer(S, A, centr=A)
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)
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_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)
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_orbits_transversals_from_bsgs():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
result = _orbits_transversals_from_bsgs(base, strong_gens_distr)
orbits = result[0]
transversals = result[1]
base_len = len(base)
for i in range(base_len):
for el in orbits[i]:
assert transversals[i][el](base[i]) == el
for j in range(i):
assert transversals[i][el](base[j]) == base[j]
order = 1
for i in range(base_len):
order *= len(orbits[i])
assert S.order() == order
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_orbits_transversals_from_bsgs():
S = SymmetricGroup(4)
S.schreier_sims()
base = S.base
strong_gens = S.strong_gens
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
result = _orbits_transversals_from_bsgs(base, strong_gens_distr)
orbits = result[0]
transversals = result[1]
base_len = len(base)
for i in range(base_len):
for el in orbits[i]:
assert transversals[i][el](base[i]) == el
for j in range(i):
assert transversals[i][el](base[j]) == base[j]
order = 1
for i in range(base_len):
order *= len(orbits[i])
assert S.order() == order
def test_centralizer():
# the centralizer of the trivial group is the entire group
S = SymmetricGroup(2)
assert S.centralizer(Permutation(list(range(2)))).is_subgroup(S)
A = AlternatingGroup(5)
assert A.centralizer(Permutation(list(range(5)))).is_subgroup(A)
# a centralizer in the trivial group is the trivial group itself
triv = PermutationGroup([Permutation([0, 1, 2, 3])])
D = DihedralGroup(4)
assert triv.centralizer(D).is_subgroup(triv)
# brute-force verifications for centralizers of groups
for i in (4, 5, 6):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
C = CyclicGroup(i)
D = DihedralGroup(i)
for gp in (S, A, C, D):
for gp2 in (S, A, C, D):
if not gp2.is_subgroup(gp):
assert _verify_centralizer(gp, gp2)
# verify the centralizer for all elements of several groups
S = SymmetricGroup(5)
elements = list(S.generate_dimino())
for element in elements:
assert _verify_centralizer(S, element)
A = AlternatingGroup(5)
elements = list(A.generate_dimino())
for element in elements:
assert _verify_centralizer(A, element)
D = DihedralGroup(7)
elements = list(D.generate_dimino())
for element in elements:
assert _verify_centralizer(D, element)
# verify centralizers of small groups within small groups
small = []
for i in (1, 2, 3):
small.append(SymmetricGroup(i))
small.append(AlternatingGroup(i))
small.append(DihedralGroup(i))
small.append(CyclicGroup(i))
for gp in small:
for gp2 in small:
if gp.degree == gp2.degree:
assert _verify_centralizer(gp, gp2)
def test_SymmetricGroup():
G = SymmetricGroup(5)
elements = list(G.generate())
assert (G.generators[0]).size == 5
assert len(elements) == 120
assert G.is_solvable is False
assert G.is_abelian is False
assert G.is_nilpotent is False
assert G.is_transitive() is True
H = SymmetricGroup(1)
assert H.order() == 1
L = SymmetricGroup(2)
assert L.order() == 2
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 test_pointwise_stabilizer():
S = SymmetricGroup(2)
stab = S.pointwise_stabilizer([0])
assert stab.generators == [Permutation(1)]
S = SymmetricGroup(5)
points = []
stab = S
for point in (2, 0, 3, 4, 1):
stab = stab.stabilizer(point)
points.append(point)
assert S.pointwise_stabilizer(points).is_subgroup(stab)
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 _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 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)
def test_cmp_perm_lists():
S = SymmetricGroup(4)
els = list(S.generate_dimino())
other = els[:]
shuffle(other)
assert _cmp_perm_lists(els, other) is True
def test_is_primitive():
S = SymmetricGroup(5)
assert S.is_primitive() is True
C = CyclicGroup(7)
assert C.is_primitive() is True
def test_naive_list_centralizer():
# verified by GAP
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert _naive_list_centralizer(S, S) == [Permutation([0, 1, 2])]
assert PermutationGroup(_naive_list_centralizer(S, A)).is_subgroup(A)
def test_cmp_perm_lists():
S = SymmetricGroup(4)
els = list(S.generate_dimino())
other = els[:]
shuffle(other)
assert _cmp_perm_lists(els, other) is True
def test_max_div():
S = SymmetricGroup(10)
assert S.max_div == 5
def test_is_primitive():
S = SymmetricGroup(5)
assert S.is_primitive() is True
C = CyclicGroup(7)
assert C.is_primitive() is True
def test_commutator():
# the commutator of the trivial group and the trivial group is trivial
S = SymmetricGroup(3)
triv = PermutationGroup([Permutation([0, 1, 2])])
assert S.commutator(triv, triv).is_subgroup(triv)
# the commutator of the trivial group and any other group is again trivial
A = AlternatingGroup(3)
assert S.commutator(triv, A).is_subgroup(triv)
# the commutator is commutative
for i in (3, 4, 5):
S = SymmetricGroup(i)
A = AlternatingGroup(i)
D = DihedralGroup(i)
assert S.commutator(A, D).is_subgroup(S.commutator(D, A))
# the commutator of an abelian group is trivial
S = SymmetricGroup(7)
A1 = AbelianGroup(2, 5)
A2 = AbelianGroup(3, 4)
triv = PermutationGroup([Permutation([0, 1, 2, 3, 4, 5, 6])])
assert S.commutator(A1, A1).is_subgroup(triv)
assert S.commutator(A2, A2).is_subgroup(triv)
# examples calculated by hand
S = SymmetricGroup(3)
A = AlternatingGroup(3)
assert S.commutator(A, S).is_subgroup(A)
def test_is_group():
assert PermutationGroup(Permutation(1, 2), Permutation(2,
4)).is_group is True
assert SymmetricGroup(4).is_group is True
