def test_len(self):
s1 = Permutation.read_cycle_form([[1,2,3,4]], 4)
s2 = Permutation.read_cycle_form([[1,2]], 4)
G = PermGroup([s1, s2])
self.assertEqual(len(G), len(G._list_elements()))
self.assertEqual(len(G), 24)
self.assertTrue(Permutation.read_cycle_form([[3,4]], 4) in G._list_elements())
def test_change_base(self):
cf = Permutation.read_cycle_form
a = cf([[2,3],[4,6],[5,8],[9,11]], 13)
b = cf([[1,2,4,7,9,3,5,6,8,10,11,12,13]], 13)
G = PermGroup([a,b])
G.change_base([1,3])
self.assertEqual(G.base[:2], [1,3])
def test_order(self):
cf = lambda x : Permutation.read_cycle_form(x,6)
g1 = cf([[1,4,5,2,3]])
g2 = cf([[1,2],[3,4]])
g3 = cf([[1,2],[6,5]])
G = PermGroup([g1,g2,g3])
self.assertEqual(G.order(), 360)
def test_coset_enumeration(self):
g1 = Permutation.read_cycle_form([[1,2,3,4]], 4)
g2 = Permutation.read_cycle_form([[1,2]], 4)
h1 = Permutation.read_cycle_form([[1,2,3]], 4)
h2 = Permutation.read_cycle_form([[1,2]], 4)
G = PermGroup([g1, g2])
H = PermGroup([h1, h2])
cosets = G._left_cosets(H)
total = 0
elements = []
for coset in cosets:
temp_eles = coset._list_elements()
elements += temp_eles
self.assertEqual(len(temp_eles),len(H))
total += len(temp_eles)
self.assertEqual(len(G), total)
self.assertEqual(sorted(G._list_elements()), sorted(elements))
cosets = G._right_cosets(H)
total = 0
elements = []
for coset in cosets:
temp_eles = coset._list_elements()
elements += temp_eles
self.assertEqual(len(temp_eles),len(H))
total += len(temp_eles)
self.assertEqual(len(G), total)
self.assertEqual(sorted(G._list_elements()), sorted(elements))
def test_change_base_redundancy(self):
cf = lambda x: Permutation.read_cycle_form(x,6)
a = cf([[1,2,3,4],[5,6]])
b = cf([[1,2]])
G = PermGroup([a,b])
G.change_base([5,6,1,2,3,4])
self.assertEqual(G.base[:4],[5,6,1,2])
def test_partition_backtrack_subgroup_two_prop_subgroup_stab(self):
cf = Permutation.read_cycle_form
a = cf([[3,2,6]], 7)
b = cf([[3,2],[6,7]], 7)
G = PermGroup([a,b])
fam_subgroup = SubgroupFamily(G)
prop_subgroup = SubgroupProperty(G)
stab = Partition([[1,3,5,7],[2,4,6]])
fam_part_stab = PartitionStabaliserFamily(stab)
prop_part_stab = PartitionStabaliserProperty(stab)
size = 7
fam = RefinementUnion([fam_part_stab, fam_subgroup])
prop = CosetPropertyUnion([prop_part_stab, prop_subgroup])
con = PartitionStackConstraint()
mods = ModifierUnion([fam, prop, con])
ls = LeonSearch(mods, size)
gens = ls.subgroup()
found = []
s7 = PermGroup([cf([[1,2]],7),cf([[1,2,3,4,5,6,7]],7)])
for ele in s7._list_elements():
if prop.property_check(ele):
found.append(ele)
self.assertEqual(len(PermGroup(gens)), len(found))
def test_partition_backtrack_subgroup_two_prop_subgroup_stab(self):
cf = Permutation.read_cycle_form
a = cf([[3,2,6]], 7)
b = cf([[3,2],[6,7]], 7)
G = PermGroup([a,b])
fam_subgroup = SubgroupFamily(G)
prop_subgroup = SubgroupProperty(G)
stab = Partition([[1,3,5,7],[2,4,6]])
fam_part_stab = PartitionStabaliserFamily(stab)
prop_part_stab = PartitionStabaliserProperty(stab)
size = 7
fam = Refinement([fam_part_stab, fam_subgroup])
prop = CosetProperty([prop_part_stab, prop_subgroup])
pbw = PBW(prop, fam, size)
gens = pbw.find_partition_backtrack_subgroup()
found = []
s7 = PermGroup([cf([[1,2]],7),cf([[1,2,3,4,5,6,7]],7)])
for ele in s7._list_elements():
if prop.check(ele):
found.append(ele)
self.assertEqual(len(PermGroup(gens)), len(found))
def test_rand_element(self):
cf = Permutation.read_cycle_form
a = cf([[1, 2], [6,7]],7)
b = cf([[1,2,3,4], [6,7]], 7)
G=PermGroup([a,b])
for _ in range(100):
g = G._rand_element()
self.assertTrue(g in G)
def _respected_partition(comm):
comm_group = Group([comm])
orbs = comm_group.orbits()
len_orb = dict()
for orb in orbs:
if len(orb) in len_orb:
len_orb[len(orb)].extend(orb)
else:
len_orb[len(orb)] = orb
return Partition(len_orb.values())
def test_fixed_base_group(self):
g1 = Permutation.read_cycle_form([[1,2,3,4,5,6]], 6)
g2 = Permutation.read_cycle_form([[1,2]], 6)
h1 = Permutation.read_cycle_form([[3,4,5,6]], 6)
h2 = Permutation.read_cycle_form([[3,4]], 6)
G = PermGroup.fixed_base_group([g1, g2], [5,4])
H = PermGroup.fixed_base_group([h1, h2], [1,2,3,4,5,6])
N = PermGroup.fixed_base_group([g1,h1], [])
self.assertEqual(G.base[:2], [5,4])
self.assertEqual(H.base[:4], [1,2,3,4])
self.contains_membership_test(G, H)
def naive_min_gen(original_group, ordering):
eles = original_group._list_elements(key = ordering_to_perm_key(ordering))
cur_gens = [eles[1]]
cur_G = Group.fixed_base_group(cur_gens, ordering)
for ele in eles:
if ele not in cur_G:
cur_gens.append(ele)
cur_G = Group.fixed_base_group(cur_gens, ordering)
if cur_G.order() == G.order():
break
return cur_gens
def test_contains(self):
cf = lambda x:Permutation.read_cycle_form(x, 6)
g1 = cf([[1,2,3,4,5,6]])
g2 = cf([[1,2]])
h1 = cf([[3,4,5,6]])
h2 = cf([[3,4]])
a1 = cf([[3,1],[6,4]])
a2 = cf([[1,6,4]])
G = PermGroup([g1, g2])
H = PermGroup([h1, h2])
self.contains_membership_test(G, H)
S6 = PermGroup.fixed_base_group([g1,g2], [5,2,1,3,4,6])
A4 = PermGroup.fixed_base_group([a1,a2], [3,1,2,5,6,4])
self.contains_membership_test(S6, A4)
def test_pre_post_element(self):
cf = Permutation.read_cycle_form
a = cf([[2,3],[4,6],[5,8],[9,11]], 13)
b = cf([[1,2,4,7,9,3,5,6,8,10,11,12,13]], 13)
G = PermGroup.fixed_base_group([a,b], [1,3])
pre = [1,3]
post = [9,3]
self.pre_post_works(pre, post, G)
def test_base_image_member(self):
h1 = Permutation.read_cycle_form([[3,4,5,6,7]], 7)
h2 = Permutation.read_cycle_form([[3,4]], 7)
H = PermGroup.fixed_base_group([h1, h2], [3,4,5,6])
self.assertEqual(H.base, [3,4,5,6])
image1 = [1,2,3,4]
image2 = [5,3,4,6]
self.assertTrue(H.base_image_member(image1) is None)
self.assertEqual(H.base_image_member(image2), Permutation.read_cycle_form([[3,5,4]],7))
def double_coset_1(self, base, image, gens, key = None):
#base change (for now just recompute base but should do the proper algorithm)
if len(image) == 0 or len(gens) == 0:
return True
G = PermGroup.fixed_base_group(gens, image[:-1])
orb = G.orbit(image[-1], stab_level = len(base) - 1, key = key)
#print("Orbit of {} is {} in {}".format(image[-1], orb, G._list_elements()))
if image[-1] == orb[0]:
return True
return False
def test_partition_backtrack_subgroup_one_prop_stab(self):
cf = Permutation.read_cycle_form
stab = Partition([[1,3,5],[2,4,6]])
fam_part_stab = PartitionStabaliserFamily(stab)
prop_part_stab = PartitionStabaliserProperty(stab)
size = 6
fam = Refinement([fam_part_stab])
prop = CosetProperty([prop_part_stab])
pbw = PBW(prop, fam, size)
gens = pbw.find_partition_backtrack_subgroup()
found = []
s6 = PermGroup([cf([[1,2]],6),cf([[1,2,3,4,5,6]],6)])
for ele in s6._list_elements():
if prop.check(ele):
found.append(ele)
self.assertEqual(len(PermGroup(gens)), len(found))
def test_partition_backtrack_subgroup_one_prop_subgroup(self):
cf = Permutation.read_cycle_form
a = cf([[1,2,3]], 5)
b = cf([[1,2],[3,4]], 5)
G = PermGroup([a,b])
fam_subgroup = SubgroupFamily(G)
prop_subgroup = SubgroupProperty(G)
fam = Refinement([fam_subgroup])
prop = CosetProperty([prop_subgroup])
pbw = PBW(prop, fam, 5)
gens = pbw.find_partition_backtrack_subgroup()
found = []
s5 = PermGroup([cf([[1,2]],5),cf([[1,2,3,4,5]],5)])
for ele in s5._list_elements():
if prop.check(ele):
found.append(ele)
self.assertEqual(len(PermGroup(gens)), len(found))
def test_orbit(self):
g1 = Permutation.read_cycle_form([[1,2,3,4]], 4)
g2 = Permutation.read_cycle_form([[1,2]], 4)
base = [3,4,1]
reverse_priority = [3,4,1,2]
G = PermGroup.fixed_base_group([g1, g2], base)
self.assertEqual(G.orbit(1), [1,2,3,4])
self.assertEqual(G.orbit(1, stab_level = 0), [1,2,3,4])
self.assertEqual(G.orbit(1, stab_level = 1), [1,2,4])
self.assertEqual(G.orbit(1, stab_level = 2), [1,2])
self.assertEqual(G.orbit(1, stab_level = 3), [1])
self.assertEqual(G.orbit(1, key = lambda x : reverse_priority[x - 1]), [3,4,1,2])
def test_partition_backtrack_subgroup_leon_paper(self):
cf = Permutation.read_cycle_form
a = cf([[2,3],[4,6],[5,8],[9,11]], 13)
b = cf([[1,2,4,7,9,3,5,6,8,10,11,12,13]], 13)
G = PermGroup([a,b])
fam_subgroup = SubgroupFamily(G)
prop_subgroup = SubgroupProperty(G)
stab = Partition([[1,3,5,7,9,11],[2,4,6,8,10,12,13]])
fam_part_stab = PartitionStabaliserFamily(stab)
prop_part_stab = PartitionStabaliserProperty(stab)
size = 13
fam = Refinement([fam_subgroup, fam_part_stab])
prop = CosetProperty([prop_subgroup, prop_part_stab])
pbw = PBW(prop, fam, size)
gens = pbw.find_partition_backtrack_subgroup()
cand_G = PermGroup(gens)
leon_gens = []
leon_gens.append(cf([[2,12],[4,10],[5,7],[6,8]],13))
leon_gens.append(cf([[2,8],[3,5],[4,6],[10,12]],13))
leon_gens.append(cf([[1,9],[2,4],[6,8],[10,12]],13))
leon_G = PermGroup(leon_gens)
third_source = []
for ele in G._list_elements():
if prop.check(ele):
third_source.append(ele)
self.assertEqual(cand_G.order(), leon_G.order())
for perm in leon_gens:
self.assertTrue(perm in G)
self.assertTrue(perm in cand_G)
def test_partition_backtrack_subgroup_one_prop_stab(self):
cf = Permutation.read_cycle_form
stab = Partition([[1,3,5],[2,4,6]])
fam_part_stab = PartitionStabaliserFamily(stab)
prop_part_stab = PartitionStabaliserProperty(stab)
size = 6
fam = RefinementUnion([fam_part_stab])
prop = CosetPropertyUnion([prop_part_stab])
con = PartitionStackConstraint()
mods = ModifierUnion([fam, prop, con])
ls = LeonSearch(mods, size)
gens = ls.subgroup()
found = []
s6 = PermGroup([cf([[1,2]],6),cf([[1,2,3,4,5,6]],6)])
for ele in s6._list_elements():
if prop.property_check(ele):
found.append(ele)
self.assertEqual(sorted(PermGroup(gens)._list_elements()), sorted(found))
def test_pre_post_element_A4(self):
cf = Permutation.read_cycle_form
a = cf([[1,2],[3,4]], 5)
b = cf([[1,2,3]], 5)
G = PermGroup.fixed_base_group([a,b], [5,4,3,2,1])
pre = [5,4]
post = [4,3]
ele = G.prefix_postfix_image_member(pre,post)
self.assertTrue(ele is None)
pre = [5,4,3]
post = [5,2,1]
self.pre_post_works(pre, post, G)
ele = G.prefix_postfix_image_member(pre,post)
self.assertEqual(cf([[1,3],[2,4]],5),ele)
def test_partition_backtrack_subgroup_one_prop_subgroup(self):
cf = Permutation.read_cycle_form
a = cf([[1,2,3]], 5)
b = cf([[1,2],[3,4]], 5)
G = PermGroup([a,b])
fam_subgroup = SubgroupFamily(G)
prop_subgroup = SubgroupProperty(G)
fam = RefinementUnion([fam_subgroup])
prop = CosetPropertyUnion([prop_subgroup])
con = PartitionStackConstraint()
mods = ModifierUnion([fam, prop, con])
ls = LeonSearch(mods, 5)
gens = ls.subgroup()
found = []
s5 = PermGroup([cf([[1,2]],5),cf([[1,2,3,4,5]],5)])
for ele in s5._list_elements():
if prop.property_check(ele):
found.append(ele)
self.assertEqual(sorted(PermGroup(gens)._list_elements()), sorted(found))
def schreier_sims0():
from permutation import Permutation
from refinement import SubgroupFamily, PartitionStabaliserFamily, Refinement
from partition import Partition
from group import PermGroup
from leon_search import PartitionBacktrackWorkhorse as PBW
from coset_property import CosetProperty, SubgroupProperty, PartitionStabaliserProperty
cf = Permutation.read_cycle_form
a = cf([[1, 2]], 4)
b = cf([[1, 2, 3, 4]], 4)
G = PermGroup.fixed_base_group([a, b], [1, 2, 3, 4])
base = G.base
graphs = G.schreier_graphs
gens = G.chain_generators
print("Base is: {}".format(base))
for i in range(len(graphs)):
print("Generators and schreier graph for G[{}]:".format(i + 1))
print(gens[i])
print(graphs[i])
from group import PermGroup
from permutation import Permutation
_base_size = 6
cf = lambda x : Permutation.read_cycle_form(x, _base_size)
a= cf([[1,2,3,4]])
b= cf([[1,2]])
c= cf([[1,2,3]])
s4 = PermGroup([a,b])
s3 = PermGroup([b,c])
for coset in s4._right_cosets(s3):
print(coset._list_elements())
for coset in s4._left_cosets(s3):
print(coset._list_elements())
cf = Permutation.read_cycle_form
a = cf([[1, 2, 3,4,5]],5)
b = cf([[1,2,3]], 5)
G=PermGroup([a,b])
print(len(G))
def test_orbits(self):
cf = Permutation.read_cycle_form
a = cf([[1, 2], [6,7]],7)
b = cf([[1,2,3,4], [6,7]], 7)
G=PermGroup.fixed_base_group([a,b], [1,2,3])
self.assertEqual(sorted(G.orbits()), [[1,2,3,4],[5],[6,7]])
from ordering import ordering_to_key, ordering_to_perm_key
#Set up group generators
cf = Permutation.read_cycle_form
a = cf([[2,3],[4,6],[5,8],[9,11]], 13)
b = cf([[1,2,4,7,9,3,5,6,8,10,11,12,13]], 13)
#Define ordering of base elements
fix = [1,3,5,9,7,11,2,12,4,6,8,10,13]
#Define corresponding key functions on numbers and perms based on base element ordering
key_single = ordering_to_key(fix)
key_perm = ordering_to_perm_key(fix)
#Construct group
G = Group.fixed_base_group([a,b],fix)
#Find all elements in order
#Print the base and stab orbits of G.
print(G.base)
level_orbits = []
for level in range(len(G.base)):
to_check = sorted(G.orbit(G.base[level],level), key = key_single)
level_orbits.append(to_check)
print(to_check)
#Set up the modulo values for a naive traversal.
mods = [len(orbit) for orbit in level_orbits]
def inc(count, resets):#incrementor function for the naive traversal
sig = len(count) - 1
while (count[sig] + 1) % resets[sig] == 0:
if __name__ == '__main__':
from _example_path_tools import add_path_examples
add_path_examples()
from permutation import Permutation
from group import PermGroup
from schreier_sims import _coset_rep_inverse as rep
cf = lambda x: Permutation.read_cycle_form(x,5)
a = cf([[2,3,4]])
b = cf([[1,2,3,4,5]])
A5 = PermGroup.fixed_base_group([a,b],[1,2,3])
print(A5.base)
print(A5.strong_generators)
for sg in A5.schreier_graphs:
for image in range(1, 6):
print("{}: {}".format(image, rep(image, sg, A5.identity)))
from _example_path_tools import add_path_examples
add_path_examples()
from partition import Partition
from permutation import Permutation
from group import PermGroup as Group
from coset_property import permutation_commuter
cf =Permutation.read_cycle_form
a = cf([[1, 2]],4)
check = permutation_commuter(a)
b = cf([[1,2,3,4]], 4)
c = cf([[3, 4]], 4)
d = cf([[2, 3]], 4)
e = cf([[3, 4]], 4)
G = Group([a,b])
G1 = Group([c,d])
G2 = Group([e])
identity = cf([],4)
sub = Group([identity])
for g in G._list_elements():
d = sub * g * sub
p_flag = check(g)
d_flag = d.mid_minimal()
if g != identity and p_flag and d_flag:
sub = Group([t for t in sub.generators + [g] if t != identity])
print("{}\nprop: {}\nmin: {} sub: {}\nd: {}\n".format(g, p_flag, d_flag, sub.generators, d._list_elements()))
Dcosets = [(G2*g*G1, g) for g in G._list_elements()]
for d, g in sorted(Dcosets,key = lambda x: x[0]._list_elements()):
eles = d._list_elements()
#print("{} ({}) {}:\n{}\n".format(g,len(eles),d.mid_minimal(),eles))
