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 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_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 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_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_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 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_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 = 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 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 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_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 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)
