def pgl2_a4_2(q, a):
'''
q is assumed to be 5 mod 6
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
two = field.one() + field.one()
three = two + field.one()
nonsquare = SquareExtensionField.get_non_square_element(field)
nonsquare_3_roots = [
x for x in field.get_all_elements() if three * x * x == -nonsquare
]
x = nonsquare_3_roots[0]
bs = [
b for b in field.get_all_elements()
if b * b - two * b * x * x + three * three * x * x * x * x +
two * two * a * a * x * x == field.zero()
]
if len(bs) == 0:
return []
s2_1 = pgl2.create([[a, bs[0]], [field.one(), -a]])
s3 = pgl2.create([[x, -three * x * x], [field.one(), x]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements
def pgl2_a4_1(q, a):
'''
q is assumed to be 1 mod 6
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
unity_3_roots = [
x for x in field.get_all_elements()
if x * x + x + field.one() == field.zero()
]
zeta = unity_3_roots[0]
field_two = field.one() + field.one()
s2_1 = pgl2.create([[a, field.one()], [field_two * a * a, -a]])
if s2_1.det() == field.zero():
return []
s3 = pgl2.create([[zeta, field.zero()], [field.zero(), field.one()]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements
def get_q_groups_from_file(filename):
result = {}
with open(filename) as file:
for group_data in file:
if group_data == '\n':
continue
first_comma = group_data.index(',')
q = int(group_data[:first_comma])
pgl = PGL(2, q)
field = pgl.get_field()
if q not in result:
result[q] = []
group_data = group_data[first_comma+4:-3]
elements = []
element_strings = group_data.split('], [')
for element_string in element_strings:
rows = element_string.split(';')
element_data = [row.split(',') for row in rows]
element_data = [[field.create_element(int(x)) for x in row] for row in element_data]
element = pgl.create(element_data)
elements.append(element)
result[q].append(set(elements))
return result
def pgl2_a4_0(q, a):
'''
q is assumed to be 3 ** k
'''
pgl2 = PGL(2, q)
field = pgl2.get_field()
identity = pgl2.identity()
s2_1 = pgl2.create([[a, -a * a - field.one()], [field.one(), -a]])
s3 = pgl2.create([[field.one(), field.one()], [field.zero(), field.one()]])
s2_2 = s3 * s2_1 * s3.inverse()
s2_3 = s2_1 * s2_2
elements = [identity, s2_1, s2_2, s2_3]
elements_addition = [x * s3 for x in elements]
elements_addition += [x * s3 * s3 for x in elements]
elements += elements_addition
return elements
e3 = pgl.create([
[one, zero, zero],
[zero, one, zero],
[extension_element, one, one]
])
e4 = pgl.create([
[one, zero, zero],
[one, one, zero],
[zero, zero, one]
])
members = {pgl.identity()}
for k in range(8):
for x in product([e1, e2, e3, e4], members):
res = x[0] * x[1]
members.add(res)
if len(members) == 360:
break
return members
pgl = PGL(3, 4)
action = PGLGroupAction(pgl)
members = a6_in_psl3_4(pgl)
vector_space = action.get_acted_upon_elements()
orbit_elements = [action.apply(g, vector_space[0]) for g in members]
orbit = list(set(orbit_elements))
print(len(orbit))
def __init__(self, pgl: PGL):
self._pgl = pgl
self._pf = pgl.get_pf()
from projective_sets.pxl2_subsets import psl2_a4_copies
from group_actions import PGLGroupAction
from projective_sets.pgl import PGL
qs = [3, 5, 7, 11, 23, 31, 47]
for q in qs:
pgl = PGL(2, q)
action = PGLGroupAction(pgl)
field = pgl.get_pf().get_field()
a4_copies = psl2_a4_copies(field)
good_a4 = [a4_copies[0]]
a4_copies.pop(0)
if q % 8 in [1, 7]:
while len(a4_copies) > 0 or len(good_a4) < 2:
a4 = a4_copies[0]
unique_a4 = True
for element in pgl.get_all_elements():
if element.det().legendre() != 1:
continue
a4_conjugated = set([element*x*element.inverse() for x in a4])
if a4_conjugated in good_a4:
unique_a4 = False
break
if unique_a4:
good_a4.append(a4)
a4_copies.pop(0)
for copy in good_a4:
print(q, copy)
from utilities.q_group_from_file import get_q_groups_from_file
from group_actions.group_utiliies import get_extension_of_order_2
from projective_sets.pgl import PGL
group_copies = get_q_groups_from_file(r'results/groups_in_low_psl2q_up_to_conjugation.txt')
good_groups = {}
for q in group_copies:
good_groups[q] = []
pgl = PGL(2, q)
for copy in group_copies[q]:
extension = get_extension_of_order_2(copy, pgl, lambda x: x.det().legendre() == -1)
good_groups[q].append(extension)
for x in good_groups[q]:
print(q, x)