class TestRubikGroup4Corner(unittest.TestCase):
def setUp(self):
self.N = 93
self.group = Group()
self.order_rubik4 = 707195371192426622240452051915172831683411968000000000
X1 = Perm()(1,74,0,34)(2,70,92,38)(3,66,91,42)\
(4,63,90,46)(47,50,62,59)(48,54,61,55)(49,58,60,51)(52,53,57,56)
X2 = Perm()(5,75,89,33)(6,71,88,37)(7,67,87,41)(8,64,86,45)
X3 = Perm()(9,76,85,32)(10,72,84,36)(11,68,83,40)(12,65,82,44)
Y1 = Perm()(4,59,81,18)(8,55,85,22)(12,51,89,26)\
(15,47,0,30)(31,34,46,43)(32,38,45,39)(33,42,44,35)(36,37,41,40)
Y2 = Perm()(3,60,80,17)(7,56,84,21)(11,52,88,25)(14,48,92,29)
Y3 = Perm()(2,61,79,16)(6,57,83,20)(10,53,87,24)(13,49,91,28)
Z1 = Perm()(27,43,59,74)(28,44,60,75)(29,45,61,76)\
(30,46,62,77)(78,81,0,90)(79,85,92,86)(80,89,91,82)(83,84,88,87)
Z2 = Perm()(23,39,55,70)(24,40,56,71)(25,41,57,72)(26,42,58,73)
Z3 = Perm()(19,35,51,66)(20,36,52,67)(21,37,53,68)(22,38,54,69)
#self.generators = [X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3]
self.generators = [X1, X2, X3] # order 64
#self.generators = [Y1, Y2, Y3] # order 64
#self.generators = [Z1, Z2, Z3] # order 64
def test_insert_generators(self):
for perm in self.generators:
self.group.insert(perm)
#self.assertEqual(self.group.order(), self.order_rubik4)
self.assertEqual(self.group.order(), 64)
def tearDown(self): pass
def test_orbits4(self):
self.N = 10
self.group = Group()
self.group.insert(Perm()(0, 1, 2))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(
self.group.is_transitive(strict=False, points=range(self.N)))
class TestSudoku4x4(unittest.TestCase):
def setUp(self):
self.N = 16
self.group = Group()
self.generators = []
self.generators.append(
Perm()(1,2)(5,6)(9,10)(13,14))
self.generators.append(
Perm()(3,4)(7,8)(11,12)(15,0))
self.generators.append(
Perm()(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,0))
self.generators.append(
Perm()(1,5)(2,6)(3,7)(4,8))
self.generators.append(
Perm()(9,13)(10,14)(11,15)(12,0))
self.generators.append(
Perm()(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,0))
self.generators.append(
Perm()(1,4,0,13)(2,8,15,9)(3,12,14,5)(6,7,11,10))
def test_insert(self):
for perm in self.generators:
self.group.insert(perm)
self.assertEqual(self.group.order(), 128)
def tearDown(self): pass
class TestRubikGroup2(unittest.TestCase):
def setUp(self):
self.N = 21
self.group = Group()
self.order_rubik2 = 3674160 # 6 * 9 * 12 * 15 * 18 * 21
R1 = Perm()(2,13,19,4) * Perm()(3,11,0,6) * Perm()(7,8,10,9)
D1 = Perm()(5,9,13,16) * Perm()(6,10,14,17) * Perm()(18,19,0,20)
B1 = Perm()(1,16,0,8) * Perm()(2,15,20,10) * Perm()(11,12,14,13)
R2 = R1 * R1
R3 = R1 * R2
D2 = D1 * D1
D3 = D1 * D2
B2 = B1 * B1
B3 = B1 * B2
self.generators = [R1, D1, B1]
# cwiartki i polowki
self.face_turns = [R1, R2, R3, D1, D2, D3, B1, B2, B3]
# tylko cwiartki
self.quarter_turns = [R1, R3, D1, D3, B1, B3]
def test_insert_generators(self):
for perm in self.generators:
#print "insert", perm
#self.group.insert(perm)
pass # too slow and to big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert_face_turns(self):
for perm in self.face_turns:
#print "insert", perm
#self.group.insert(perm)
pass # too slow and to big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert_quarter_turns(self):
for perm in self.quarter_turns:
#print "insert", perm
#self.group.insert(perm)
pass # too slow and to big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert(self):
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2)(3, 5, 4))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(0, 3)(1, 4)(2, 5))
self.assertEqual(self.group.order(), 6)
self.group.insert(Perm()(0, 6)(1, 7)(2, 8))
self.assertEqual(self.group.order(), 6 * 9)
#self.group.insert(Perm()(0, 9)(1, 10)(2, 11))
#self.assertEqual(self.group.order(), 6 * 9 * 12)
#self.group.insert(Perm()(0, 12)(1, 13)(2, 14))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15)
#self.group.insert(Perm()(0, 15)(1, 16)(2, 17))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15 * 18)
#self.group.insert(Perm()(0, 18)(1, 19)(2, 20))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15 * 18 * 21)
def tearDown(self): pass
class TestRubikGroup2(unittest.TestCase):
def setUp(self):
self.N = 21
self.group = Group()
self.order_rubik2 = 3674160 # 6 * 9 * 12 * 15 * 18 * 21
R1 = Perm()(2, 13, 19, 4) * Perm()(3, 11, 0, 6) * Perm()(7, 8, 10, 9)
D1 = Perm()(5, 9, 13, 16) * Perm()(6, 10, 14, 17) * Perm()(18, 19, 0,
20)
B1 = Perm()(1, 16, 0, 8) * Perm()(2, 15, 20, 10) * Perm()(11, 12, 14,
13)
R2 = R1 * R1
R3 = R1 * R2
D2 = D1 * D1
D3 = D1 * D2
B2 = B1 * B1
B3 = B1 * B2
self.generators = [R1, D1, B1]
# cwiartki i polowki
self.face_turns = [R1, R2, R3, D1, D2, D3, B1, B2, B3]
# tylko cwiartki
self.quarter_turns = [R1, R3, D1, D3, B1, B3]
def test_insert_generators(self):
for perm in self.generators:
#self.group.insert(perm)
pass # too slow and too big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert_face_turns(self):
for perm in self.face_turns:
#self.group.insert(perm)
pass # too slow and too big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert_quarter_turns(self):
for perm in self.quarter_turns:
#self.group.insert(perm)
pass # too slow and too big
#self.assertEqual(self.group.order(), self.order_rubik2)
def test_insert(self):
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2)(3, 5, 4))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(0, 3)(1, 4)(2, 5))
self.assertEqual(self.group.order(), 6)
self.group.insert(Perm()(0, 6)(1, 7)(2, 8))
self.assertEqual(self.group.order(), 6 * 9)
#self.group.insert(Perm()(0, 9)(1, 10)(2, 11))
#self.assertEqual(self.group.order(), 6 * 9 * 12)
#self.group.insert(Perm()(0, 12)(1, 13)(2, 14))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15)
#self.group.insert(Perm()(0, 15)(1, 16)(2, 17))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15 * 18)
#self.group.insert(Perm()(0, 18)(1, 19)(2, 20))
#self.assertEqual(self.group.order(), 6 * 9 * 12 * 15 * 18 * 21)
def tearDown(self):
pass
def test_is_subgroup(self):
self.group2 = Group()
# Tworze grupe cykliczna C_N.
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
self.assertTrue(self.group2.is_subgroup(self.group1))
self.assertTrue(self.group2.is_abelian())
self.assertFalse(self.group1.is_abelian())
self.assertFalse(self.group2.is_normal(self.group1))
def test_orbits1(self):
self.N = 3
self.group = Group()
self.group.insert(Perm()(0, 1))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1], [2]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(
self.group.is_transitive(points=range(self.N), strict=False))
def test_orbits2(self):
self.N = 4
self.group = Group()
self.group.insert(Perm()(0, 1))
self.group.insert(Perm()(2, 3))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1], [2, 3]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertEqual(self.group.orbits([0, 1, 2]), [[0, 1], [2, 3]])
def test_centralizer(self):
# Tworze grupe cykliczna.
self.group2 = Group()
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
# centrum grupy abelowej cyklicznej to cala grupa
self.group3 = self.group2.center()
self.assertEqual(self.group3.order(), self.N)
# Dalej dla grupy symetrycznej.
self.group2 = self.group1.center()
self.assertEqual(self.group2.order(), 1)
def test_alt_n(self):
self.N = 5
self.assertTrue(self.N > 2)
self.group = Group()
order = 1
for i in range(self.N - 2):
self.group.insert(Perm()(i, i + 1, i + 2))
order = order * (i + 3)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def setUp(self):
self.N = 6
self.assertTrue(self.N > 2)
self.group = Group()
self.H = Perm()(*range(self.N))
self.group.insert(self.H)
left = 1
right = self.N - 1
perm = Perm()
while left < right:
perm = perm * Perm()(left, right)
left = left + 1
right = right - 1
self.group.insert(perm)
def test_alt_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(1, 2, 3))
self.assertEqual(self.group.order(), 12)
self.group.insert(Perm()(2, 3, 4))
self.assertEqual(self.group.order(), 60)
self.assertFalse(Perm()(0, 1) in self.group)
self.assertTrue(Perm()(*range(self.N)) in self.group) # N is odd
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_cyclic6(self):
points = range(5)
C6 = Group()
C6.insert(Perm()(0, 1, 2)(3, 4))
self.assertEqual(C6.order(), 6)
self.assertEqual(C6.orbits(points), [[0, 2, 1], [3, 4]])
self.assertEqual(len(C6.orbits(points)), 2)
self.assertFalse(C6.is_transitive(points))
def setUp(self):
self.N = 48
self.group = Group()
self.order_rubik3 = 43252003274489856000
U1 = Perm()(1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19)
L1 = Perm()(33,35,40,38)(34,37,39,36)(1,9,41,32)(4,12,44,29)(6,14,46,27)
F1 = Perm()(9,11,16,14)(10,13,15,12)(6,17,43,40)(7,20,42,37)(8,22,41,35)
R1 = Perm()(17,19,24,22)(18,21,23,20)(8,25,0,16)(5,28,45,13)(3,30,43,11)
B1 = Perm()(25,27,32,30)(26,29,31,28)(3,33,46,24)(2,36,47,21)(1,38,0,19)
D1 = Perm()(41,43,0,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)
U2 = U1 * U1
U3 = U1 * U2
L2 = L1 * L1
L3 = L1 * L2
F2 = F1 * F1
F3 = F1 * F2
R2 = R1 * R1
R3 = R1 * R2
D2 = D1 * D1
D3 = D1 * D2
B2 = B1 * B1
B3 = B1 * B2
#self.generators = [U1, L1, F1, R1, B1, D1]
self.generators = [U1, D1] # order 16
# cwiartki i polowki
#self.face_turns = [U1, U2, U3, L1, L2, L3, F1, F2, F3, R1, R2, R3, D1, D2, D3, B1, B2, B3]
self.face_turns = [L1, R1] # order 16
# tylko cwiartki
#self.quarter_turns = [U1, U3, L1, L3, F1, F3, R1, R3, D1, D3, B1, B3]
self.quarter_turns = [F1, B1] # order 16
def test_orbits1(self):
self.N = 3
self.group = Group()
self.group.insert(Perm()(0,1))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1],[2]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N), strict=False))
def test_is_subgroup(self):
self.group2 = Group()
# Tworze grupe cykliczna.
self.group2.insert(Perm()(*range(self.N)))
self.assertTrue(self.group2.is_subgroup(self.group1))
self.assertTrue(self.group2.is_abelian())
self.assertFalse(self.group1.is_abelian())
self.assertFalse(self.group2.is_normal(self.group1))
class TestCyclicGroup(unittest.TestCase):
def setUp(self):
# The cyclic group from the paper by Knuth.
self.N = 6
self.H = Perm()(0, 1, 2, 4)(3, 5)
self.group = Group()
self.group.insert(self.H)
def test_group(self):
self.assertEqual(self.group.order(), 4)
self.assertFalse(self.group.is_trivial())
self.assertTrue(Perm() in self.group)
self.assertTrue(self.H in self.group)
points = range(self.N)
self.assertFalse(Perm()(*points) in self.group)
self.assertEqual(self.group.orbits(points), [[0, 4, 2, 1], [3, 5]])
self.assertEqual(len(self.group.orbits(points)), 2)
self.assertFalse(self.group.is_transitive(points))
def test_cyclic6(self):
points = range(5)
C6 = Group()
C6.insert(Perm()(0, 1, 2)(3, 4))
self.assertEqual(C6.order(), 6)
self.assertEqual(C6.orbits(points), [[0, 2, 1], [3, 4]])
self.assertEqual(len(C6.orbits(points)), 2)
self.assertFalse(C6.is_transitive(points))
def tearDown(self): pass
class TestCyclicGroup(unittest.TestCase):
def setUp(self):
# The cyclic group from the paper by Knuth.
self.N = 6
self.H = Perm()(0, 1, 2, 4)(3, 5)
self.group = Group()
self.group.insert(self.H)
def test_group(self):
self.assertEqual(self.group.order(), 4)
self.assertFalse(self.group.is_trivial())
self.assertTrue(Perm() in self.group)
self.assertTrue(self.H in self.group)
points = range(self.N)
self.assertFalse(Perm()(*points) in self.group)
self.assertEqual(self.group.orbits(points), [[0, 4, 2, 1], [3, 5]])
self.assertEqual(len(self.group.orbits(points)), 2)
self.assertFalse(self.group.is_transitive(points))
def test_cyclic6(self):
points = range(5)
C6 = Group()
C6.insert(Perm()(0, 1, 2)(3, 4))
self.assertEqual(C6.order(), 6)
self.assertEqual(C6.orbits(points), [[0, 2, 1], [3, 4]])
self.assertEqual(len(C6.orbits(points)), 2)
self.assertFalse(C6.is_transitive(points))
def tearDown(self):
pass
def test_orbits2(self):
self.N = 4
self.group = Group()
self.group.insert(Perm()(0, 1))
self.group.insert(Perm()(2, 3))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1],[2, 3]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertEqual(self.group.orbits([0, 1, 2]), [[0, 1],[2, 3]])
def test_cyclic6(self):
points = range(5)
C6 = Group()
C6.insert(Perm()(0, 1, 2)(3, 4))
self.assertEqual(C6.order(), 6)
self.assertEqual(C6.orbits(points), [[0, 2, 1], [3, 4]])
self.assertEqual(len(C6.orbits(points)), 2)
self.assertFalse(C6.is_transitive(points))
def test_sym_n(self):
self.N = 4
self.group = Group()
order = 1
for i in range(self.N-1):
self.group.insert(Perm()(i, i+1))
order *= (i+2)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_alt_n(self):
self.N = 5
self.assertTrue(self.N > 2)
self.group = Group()
order = 1
for i in range(self.N-2):
self.group.insert(Perm()(i, i+1, i+2))
order = order * (i+3)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_centralizer(self):
# Tworze grupe cykliczna.
self.group2 = Group()
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
# centrum grupy abelowej cyklicznej to cala grupa
self.group3 = self.group2.center()
self.assertEqual(self.group3.order(), self.N)
# Dalej dla grupy symetrycznej.
self.group2 = self.group1.center()
self.assertEqual(self.group2.order(), 1)
def setUp(self):
self.N = 21
self.group = Group()
self.order_rubik2 = 3674160 # 6 * 9 * 12 * 15 * 18 * 21
R1 = Perm()(2, 13, 19, 4) * Perm()(3, 11, 0, 6) * Perm()(7, 8, 10, 9)
D1 = Perm()(5, 9, 13, 16) * Perm()(6, 10, 14, 17) * Perm()(18, 19, 0,
20)
B1 = Perm()(1, 16, 0, 8) * Perm()(2, 15, 20, 10) * Perm()(11, 12, 14,
13)
R2 = R1 * R1
R3 = R1 * R2
D2 = D1 * D1
D3 = D1 * D2
B2 = B1 * B1
B3 = B1 * B2
self.generators = [R1, D1, B1]
# cwiartki i polowki
self.face_turns = [R1, R2, R3, D1, D2, D3, B1, B2, B3]
# tylko cwiartki
self.quarter_turns = [R1, R3, D1, D3, B1, B3]
def test_alt_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(1, 2, 3))
self.assertEqual(self.group.order(), 12)
self.group.insert(Perm()(2, 3, 4))
self.assertEqual(self.group.order(), 60)
self.assertFalse(Perm()(0, 1) in self.group)
self.assertTrue(Perm()(*range(self.N)) in self.group) # N is odd
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
class TestDihedralGroup(unittest.TestCase):
def setUp(self):
self.N = 6
self.assertTrue(self.N > 2)
self.group = Group()
self.H = Perm()(*range(self.N))
self.group.insert(self.H)
left = 1
right = self.N - 1
perm = Perm()
while left < right:
perm = perm * Perm()(left, right)
left = left + 1
right = right - 1
self.group.insert(perm)
def test_insert(self):
self.assertEqual(self.group.order(), 2 * self.N)
self.assertTrue(Perm() in self.group)
self.assertTrue(self.H in self.group)
def tearDown(self):
pass
def setUp(self):
self.N = 51
self.group = Group()
self.order_rubik3 = 43252003274489856000
X1 = Perm()(1,40,0,19)(2,37,50,22)(3,35,49,25)(26,28,34,32)(27,31,33,29)
X2 = Perm()(4,41,48,18)(5,38,47,21)(6,36,46,24)
Y1 = Perm()(3,32,45,10)(6,29,48,13)(8,26,0,16)(17,19,25,23)(18,22,24,20)
Y2 = Perm()(2,33,44,9)(5,30,47,12)(7,27,50,15)
Z1 = Perm()(14,23,32,40)(15,24,33,41)(16,25,34,42)(43,45,0,49)(44,48,50,46)
Z2 = Perm()(11,20,29,37)(12,21,30,38)(13,22,31,39)
#self.generators = [X1, X2, Y1, Y2, Z1, Z2]
#self.generators = [X1, X2] # order 16
#self.generators = [Y1, Y2] # order 16
self.generators = [Z1, Z2] # order 16
class TestRubikGroup3Corner(unittest.TestCase):
def setUp(self):
self.N = 51
self.group = Group()
self.order_rubik3 = 43252003274489856000
X1 = Perm()(1,40,0,19)(2,37,50,22)(3,35,49,25)(26,28,34,32)(27,31,33,29)
X2 = Perm()(4,41,48,18)(5,38,47,21)(6,36,46,24)
Y1 = Perm()(3,32,45,10)(6,29,48,13)(8,26,0,16)(17,19,25,23)(18,22,24,20)
Y2 = Perm()(2,33,44,9)(5,30,47,12)(7,27,50,15)
Z1 = Perm()(14,23,32,40)(15,24,33,41)(16,25,34,42)(43,45,0,49)(44,48,50,46)
Z2 = Perm()(11,20,29,37)(12,21,30,38)(13,22,31,39)
#self.generators = [X1, X2, Y1, Y2, Z1, Z2]
#self.generators = [X1, X2] # order 16
#self.generators = [Y1, Y2] # order 16
self.generators = [Z1, Z2] # order 16
def test_insert_generators(self):
for perm in self.generators:
self.group.insert(perm)
#self.assertEqual(self.group.order(), self.order_rubik3)
self.assertEqual(self.group.order(), 16)
def tearDown(self): pass
def test_sym_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1))
self.assertEqual(self.group.order(), 2)
self.group.insert(Perm()(1, 2))
self.assertEqual(self.group.order(), 6)
self.group.insert(Perm()(2, 3))
self.assertEqual(self.group.order(), 24)
self.group.insert(Perm()(3, 4))
self.assertEqual(self.group.order(), 120)
self.assertTrue(Perm()(*range(self.N)) in self.group)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def setUp(self):
self.N = 16
self.group = Group()
self.generators = []
self.generators.append(
Perm()(1,2)(5,6)(9,10)(13,14))
self.generators.append(
Perm()(3,4)(7,8)(11,12)(15,0))
self.generators.append(
Perm()(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,0))
self.generators.append(
Perm()(1,5)(2,6)(3,7)(4,8))
self.generators.append(
Perm()(9,13)(10,14)(11,15)(12,0))
self.generators.append(
Perm()(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,0))
self.generators.append(
Perm()(1,4,0,13)(2,8,15,9)(3,12,14,5)(6,7,11,10))
def setUp(self):
self.N = 21
self.group = Group()
self.order_rubik2 = 3674160 # 6 * 9 * 12 * 15 * 18 * 21
R1 = Perm()(2,13,19,4) * Perm()(3,11,0,6) * Perm()(7,8,10,9)
D1 = Perm()(5,9,13,16) * Perm()(6,10,14,17) * Perm()(18,19,0,20)
B1 = Perm()(1,16,0,8) * Perm()(2,15,20,10) * Perm()(11,12,14,13)
R2 = R1 * R1
R3 = R1 * R2
D2 = D1 * D1
D3 = D1 * D2
B2 = B1 * B1
B3 = B1 * B2
self.generators = [R1, D1, B1]
# cwiartki i polowki
self.face_turns = [R1, R2, R3, D1, D2, D3, B1, B2, B3]
# tylko cwiartki
self.quarter_turns = [R1, R3, D1, D3, B1, B3]
class TestCyclicGroup(unittest.TestCase):
def setUp(self):
# The cyclic group from the paper by Knuth.
self.N = 6
self.H = Perm()(0, 1, 2, 4)(3, 5)
self.group = Group()
self.group.insert(self.H)
def test_group(self):
self.assertEqual(self.group.order(), 4)
self.assertFalse(self.group.is_trivial())
self.assertTrue(Perm() in self.group)
self.assertTrue(self.H in self.group)
self.assertFalse(Perm()(*range(self.N)) in self.group)
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 2)
def tearDown(self): pass
def setUp(self):
self.N = 4
# Tworze grupe symetryczna.
self.group1 = Group()
self.group1.insert(Perm()(0, 1))
self.group1.insert(Perm()(*range(self.N)))
def test_orbits4(self):
self.N = 10
self.group = Group()
self.group.insert(Perm()(0, 1, 2))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(self.group.is_transitive(strict=False, points=range(self.N)))
def setUp(self):
# The cyclic group from the paper by Knuth.
self.N = 6
self.H = Perm()(0, 1, 2, 4)(3, 5)
self.group = Group()
self.group.insert(self.H)
class TestAlternatingGroup(unittest.TestCase):
def setUp(self):
pass
def test_alt_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(1, 2, 3))
self.assertEqual(self.group.order(), 12)
self.group.insert(Perm()(2, 3, 4))
self.assertEqual(self.group.order(), 60)
self.assertFalse(Perm()(0, 1) in self.group)
self.assertTrue(Perm()(*range(self.N)) in self.group) # N is odd
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_alt_n(self):
self.N = 5
self.assertTrue(self.N > 2)
self.group = Group()
order = 1
for i in range(self.N - 2):
self.group.insert(Perm()(i, i + 1, i + 2))
order = order * (i + 3)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def tearDown(self):
pass
def setUp(self):
# The cyclic group from the paper by Knuth.
self.N = 6
self.H = Perm()(0, 1, 2, 4)(3, 5)
self.group = Group()
self.group.insert(self.H)
def test_orbits3(self): # grupa cykliczna
self.N = 10
self.group = Group()
self.group.insert(Perm()(*range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N)))
def test_is_normal(self):
a = Perm()(0, 1, 2)
b = Perm()(0, 1)
c = Perm()(0, 2, 1)
G = Group()
G.insert(a)
G.insert(b)
self.assertEqual(G.order(), 6) # G = S_3
H = Group()
H.insert(a)
H.insert(c)
self.assertEqual(H.order(), 3) # H = A_3
self.assertTrue(H.is_normal(G))
class TestSymmetricGroup(unittest.TestCase):
def setUp(self): pass
def test_sym_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1))
self.assertEqual(self.group.order(), 2)
self.group.insert(Perm()(1, 2))
self.assertEqual(self.group.order(), 6)
self.group.insert(Perm()(2, 3))
self.assertEqual(self.group.order(), 24)
self.group.insert(Perm()(3, 4))
self.assertEqual(self.group.order(), 120)
self.assertTrue(Perm()(*range(self.N)) in self.group)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_sym_n(self):
self.N = 4
self.group = Group()
order = 1
for i in range(self.N-1):
self.group.insert(Perm()(i, i+1))
order *= (i+2)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def tearDown(self): pass
def test_orbits3(self): # grupa cykliczna
self.N = 10
self.group = Group()
self.group.insert(Perm()(*range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N)))
class TestGroupOrbits(unittest.TestCase):
def setUp(self):
pass
def test_orbits1(self):
self.N = 3
self.group = Group()
self.group.insert(Perm()(0, 1))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1], [2]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(
self.group.is_transitive(points=range(self.N), strict=False))
def test_orbits2(self):
self.N = 4
self.group = Group()
self.group.insert(Perm()(0, 1))
self.group.insert(Perm()(2, 3))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1], [2, 3]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertEqual(self.group.orbits([0, 1, 2]), [[0, 1], [2, 3]])
def test_orbits3(self): # grupa cykliczna
self.N = 10
self.group = Group()
self.group.insert(Perm()(*range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N)))
def test_orbits4(self):
self.N = 10
self.group = Group()
self.group.insert(Perm()(0, 1, 2))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(
self.group.is_transitive(strict=False, points=range(self.N)))
def tearDown(self):
pass
class TestGroupOrbits(unittest.TestCase):
def setUp(self): pass
def test_orbits1(self):
self.N = 3
self.group = Group()
self.group.insert(Perm()(0,1))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1],[2]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N), strict=False))
def test_orbits2(self):
self.N = 4
self.group = Group()
self.group.insert(Perm()(0, 1))
self.group.insert(Perm()(2, 3))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertEqual(self.group.orbits(range(self.N)), [[0, 1],[2, 3]])
self.assertEqual(self.group.orbits([0, 1]), [[0, 1]])
self.assertEqual(self.group.orbits([0, 1, 2]), [[0, 1],[2, 3]])
def test_orbits3(self): # grupa cykliczna
self.N = 10
self.group = Group()
self.group.insert(Perm()(*range(self.N)))
self.assertTrue(self.group.is_transitive(points=range(self.N)))
def test_orbits4(self):
self.N = 10
self.group = Group()
self.group.insert(Perm()(0, 1, 2))
self.assertFalse(self.group.is_transitive(points=range(self.N)))
self.assertTrue(self.group.is_transitive(strict=False, points=range(self.N)))
def tearDown(self): pass
def test_normal_closure(self):
n = 5
# Make Cyclic(5).
C5 = Group()
C5.insert(Perm()(*range(n)))
self.assertEqual(C5.order(), n)
# Make Sym(5).
S5 = Group()
S5.insert(Perm()(*range(n)))
S5.insert(Perm()(0, 1))
self.assertEqual(S5.order(), 120)
A5 = S5.normal_closure(C5) # we get Alt(5)
self.assertEqual(A5.order(), 60)
for perm in A5:
self.assertTrue(perm.is_even())
def setUp(self):
self.N = 4
# Tworze grupe symetryczna.
self.group1 = Group()
self.group1.insert(Perm()(0, 1))
self.group1.insert(Perm()(*range(self.N)))
class TestSubgroup(unittest.TestCase):
def setUp(self):
self.N = 4
# Tworze grupe symetryczna.
self.group1 = Group()
self.group1.insert(Perm()(0, 1))
self.group1.insert(Perm()(*range(self.N)))
def test_subgroup_search(self):
self.assertEqual(self.group1.order(), 24)
# Dopuszczam permutacje parzyste - grupa alternujaca.
self.group2 = self.group1.subgroup_search(lambda x: x.is_even())
self.assertEqual(self.group2.order(), 12)
self.assertTrue(self.group2.is_transitive(points=range(self.N)))
#print self.group2
def test_stabilizer(self):
self.group2 = self.group1.stabilizer(3)
self.assertEqual(self.group2.order(), 6)
#print self.group2
def test_centralizer(self):
# Tworze grupe cykliczna.
self.group2 = Group()
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
# centrum grupy abelowej cyklicznej to cala grupa
self.group3 = self.group2.center()
self.assertEqual(self.group3.order(), self.N)
# Dalej dla grupy symetrycznej.
self.group2 = self.group1.center()
self.assertEqual(self.group2.order(), 1)
def test_is_subgroup(self):
self.group2 = Group()
# Tworze grupe cykliczna.
self.group2.insert(Perm()(*range(self.N)))
self.assertTrue(self.group2.is_subgroup(self.group1))
self.assertTrue(self.group2.is_abelian())
self.assertFalse(self.group1.is_abelian())
self.assertFalse(self.group2.is_normal(self.group1))
def tearDown(self): pass
class TestSubgroup(unittest.TestCase):
def setUp(self):
self.N = 4
# Tworze grupe symetryczna.
self.group1 = Group()
self.group1.insert(Perm()(0, 1))
self.group1.insert(Perm()(*range(self.N)))
def test_subgroup_search(self):
self.assertEqual(self.group1.order(), 24)
# Dopuszczam permutacje parzyste - grupa alternujaca.
self.group2 = self.group1.subgroup_search(lambda x: x.is_even())
self.assertEqual(self.group2.order(), 12)
self.assertTrue(self.group2.is_transitive(points=range(self.N)))
#print self.group2
def test_stabilizer(self):
self.group2 = self.group1.stabilizer(3)
self.assertEqual(self.group2.order(), 6)
#print self.group2
def test_centralizer(self):
# Tworze grupe cykliczna.
self.group2 = Group()
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
# centrum grupy abelowej cyklicznej to cala grupa
self.group3 = self.group2.center()
self.assertEqual(self.group3.order(), self.N)
# Dalej dla grupy symetrycznej.
self.group2 = self.group1.center()
self.assertEqual(self.group2.order(), 1)
def test_normalizer(self):
pass
def test_normal_closure(self):
n = 5
# Make Cyclic(5).
C5 = Group()
C5.insert(Perm()(*range(n)))
self.assertEqual(C5.order(), n)
# Make Sym(5).
S5 = Group()
S5.insert(Perm()(*range(n)))
S5.insert(Perm()(0, 1))
self.assertEqual(S5.order(), 120)
A5 = S5.normal_closure(C5) # we get Alt(5)
self.assertEqual(A5.order(), 60)
for perm in A5:
self.assertTrue(perm.is_even())
def test_derived_subgroup(self):
pass
def test_is_subgroup(self):
self.group2 = Group()
# Tworze grupe cykliczna C_N.
self.group2.insert(Perm()(*range(self.N)))
self.assertEqual(self.group2.order(), self.N)
self.assertTrue(self.group2.is_subgroup(self.group1))
self.assertTrue(self.group2.is_abelian())
self.assertFalse(self.group1.is_abelian())
self.assertFalse(self.group2.is_normal(self.group1))
def test_is_normal(self):
a = Perm()(0, 1, 2)
b = Perm()(0, 1)
c = Perm()(0, 2, 1)
G = Group()
G.insert(a)
G.insert(b)
self.assertEqual(G.order(), 6) # G = S_3
H = Group()
H.insert(a)
H.insert(c)
self.assertEqual(H.order(), 3) # H = A_3
self.assertTrue(H.is_normal(G))
def tearDown(self):
pass
class TestAlternatingGroup(unittest.TestCase):
def setUp(self): pass
def test_alt_5(self):
self.N = 5
self.group = Group()
self.assertEqual(self.group.order(), 1)
self.group.insert(Perm()(0, 1, 2))
self.assertEqual(self.group.order(), 3)
self.group.insert(Perm()(1, 2, 3))
self.assertEqual(self.group.order(), 12)
self.group.insert(Perm()(2, 3, 4))
self.assertEqual(self.group.order(), 60)
self.assertFalse(Perm()(0, 1) in self.group)
self.assertTrue(Perm()(*range(self.N)) in self.group) # N is odd
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def test_alt_n(self):
self.N = 5
self.assertTrue(self.N > 2)
self.group = Group()
order = 1
for i in range(self.N-2):
self.group.insert(Perm()(i, i+1, i+2))
order = order * (i+3)
self.assertEqual(self.group.order(), order)
self.assertTrue(self.group.is_transitive(points=range(self.N)))
self.assertEqual(len(self.group.orbits(range(self.N))), 1)
def tearDown(self): pass