def get_transversals(base, gens):
"""
Return transversals for the group with BSGS base, gens
"""
if not base:
return []
stabs = _distribute_gens_by_base(base, gens)
orbits, transversals = _orbits_transversals_from_bsgs(base, stabs)
transversals = [dict((x, h._array_form) for x, h in y.items()) for y in transversals]
return transversals
def test_distribute_gens_by_base():
base = [0, 1, 2]
gens = [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]),
Permutation([0, 2, 3, 1]), Permutation([3, 2, 1, 0])]
assert _distribute_gens_by_base(base, gens) == [gens,
[Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2]),
Permutation([0, 2, 3, 1])],
[Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2])]]
def get_transversals(base, gens):
"""
Return transversals for the group with BSGS base, gens
"""
if not base:
return []
stabs = _distribute_gens_by_base(base, gens)
orbits, transversals = _orbits_transversals_from_bsgs(base, stabs)
transversals = [
dict((x, h._array_form) for x, h in y.items()) for y in transversals
]
return transversals
def test_distribute_gens_by_base():
base = [0, 1, 2]
gens = [
Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2]),
Permutation([0, 2, 3, 1]),
Permutation([3, 2, 1, 0])
]
assert _distribute_gens_by_base(base, gens) == [
gens,
[
Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2]),
Permutation([0, 2, 3, 1])
], [Permutation([0, 1, 2, 3]),
Permutation([0, 1, 3, 2])]
]
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_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_handle_precomputed_bsgs():
A = AlternatingGroup(5)
A.schreier_sims()
base = A.base
strong_gens = A.strong_gens
result = _handle_precomputed_bsgs(base, strong_gens)
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
assert strong_gens_distr == result[2]
transversals = result[0]
orbits = 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 A.order() == order
def test_handle_precomputed_bsgs():
A = AlternatingGroup(5)
A.schreier_sims()
base = A.base
strong_gens = A.strong_gens
result = _handle_precomputed_bsgs(base, strong_gens)
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
assert strong_gens_distr == result[2]
transversals = result[0]
orbits = 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 A.order() == order
def _verify_bsgs(group, base, gens):
"""
Verify the correctness of a base and strong generating set.
Explanation
===========
This is a naive implementation using the definition of a base and a strong
generating set relative to it. There are other procedures for
verifying a base and strong generating set, but this one will
serve for more robust testing.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> _verify_bsgs(A, A.base, A.strong_gens)
True
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims
"""
from sympy.combinatorics.perm_groups import PermutationGroup
strong_gens_distr = _distribute_gens_by_base(base, gens)
current_stabilizer = group
for i in range(len(base)):
candidate = PermutationGroup(strong_gens_distr[i])
if current_stabilizer.order() != candidate.order():
return False
current_stabilizer = current_stabilizer.stabilizer(base[i])
if current_stabilizer.order() != 1:
return False
return True
def _verify_bsgs(group, base, gens):
"""
Verify the correctness of a base and strong generating set.
This is a naive implementation using the definition of a base and a strong
generating set relative to it. There are other procedures for
verifying a base and strong generating set, but this one will
serve for more robust testing.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> _verify_bsgs(A, A.base, A.strong_gens)
True
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims
"""
from sympy.combinatorics.perm_groups import PermutationGroup
strong_gens_distr = _distribute_gens_by_base(base, gens)
current_stabilizer = group
for i in range(len(base)):
candidate = PermutationGroup(strong_gens_distr[i])
if current_stabilizer.order() != candidate.order():
return False
current_stabilizer = current_stabilizer.stabilizer(base[i])
if current_stabilizer.order() != 1:
return False
return True
