def test_order():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x * y * x**-1 * y])
assert f.order() == 8
f = FpGroup(F, [x * y * x**-1 * y**-1, y**2])
assert f.order() is S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(
F,
[
a**250,
b**2,
c * b * c**-1 * b,
c**4,
c**-1 * a**-1 * c * a,
a**-1 * b**-1 * a * b,
],
)
assert f.order() == 2000
F, x = free_group("x")
f = FpGroup(F, [])
assert f.order() is S.Infinity
f = FpGroup(free_group("")[0], [])
assert f.order() == 1
def test_order():
from sympy import S
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() == S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
def test_order():
from sympy import S
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() == S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
def test_homomorphism():
F, a, b = free_group("a, b")
G = FpGroup(F, [a**3, b**3, (a * b)**2])
c = Permutation(3)(0, 1, 2)
d = Permutation(3)(1, 2, 3)
A = AlternatingGroup(4)
T = homomorphism(G, A, [a, b], [c, d])
assert T(a * b**2 * a**-1) == c * d**2 * c**-1
assert T.is_isomorphism()
assert T(T.invert(Permutation(3)(0, 2, 3))) == Permutation(3)(0, 2, 3)
T = homomorphism(G, AlternatingGroup(4), G.generators)
assert T.is_trivial()
assert T.kernel().order() == G.order()
F, a = free_group("a")
G = FpGroup(F, [a**8])
P = PermutationGroup([Permutation(0, 1, 2, 3), Permutation(0, 2)])
T = homomorphism(G, P, [a], [Permutation(0, 1, 2, 3)])
assert T.image().order() == 4
assert T(T.invert(Permutation(0, 2)(1, 3))) == Permutation(0, 2)(1, 3)
T = homomorphism(F, AlternatingGroup(4), F.generators, [c])
assert T.invert(c**2) == a**2
def test_order():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
assert f.order() == 8
f = FpGroup(F, [x*y*x**-1*y**-1, y**2])
assert f.order() == S.Infinity
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**250, b**2, c*b*c**-1*b, c**4, c**-1*a**-1*c*a, a**-1*b**-1*a*b])
assert f.order() == 2000
F, x = free_group("x")
f = FpGroup(F, [])
assert f.order() == S.Infinity
f = FpGroup(free_group('')[0], [])
assert f.order() == 1
def test_isomorphisms():
F, a, b = free_group("a, b")
E, c, d = free_group("c, d")
# Infinite groups with differently ordered relators.
G = FpGroup(F, [a**2, b**3])
H = FpGroup(F, [b**3, a**2])
assert is_isomorphic(G, H)
# Trivial Case
# FpGroup -> FpGroup
H = FpGroup(F, [a**3, b**3, (a*b)**2])
F, c, d = free_group("c, d")
G = FpGroup(F, [c**3, d**3, (c*d)**2])
check, T = group_isomorphism(G, H)
assert check
T(c**3*d**2) == a**3*b**2
# FpGroup -> PermutationGroup
# FpGroup is converted to the equivalent isomorphic group.
F, a, b = free_group("a, b")
G = FpGroup(F, [a**3, b**3, (a*b)**2])
H = AlternatingGroup(4)
check, T = group_isomorphism(G, H)
assert check
assert T(b*a*b**-1*a**-1*b**-1) == Permutation(0, 2, 3)
assert T(b*a*b*a**-1*b**-1) == Permutation(0, 3, 2)
# PermutationGroup -> PermutationGroup
D = DihedralGroup(8)
p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
P = PermutationGroup(p)
assert not is_isomorphic(D, P)
A = CyclicGroup(5)
B = CyclicGroup(7)
assert not is_isomorphic(A, B)
# Two groups of the same prime order are isomorphic to each other.
G = FpGroup(F, [a, b**5])
H = CyclicGroup(5)
assert G.order() == H.order()
assert is_isomorphic(G, H)
def test_isomorphisms():
F, a, b = free_group("a, b")
E, c, d = free_group("c, d")
# Infinite groups with differently ordered relators.
G = FpGroup(F, [a**2, b**3])
H = FpGroup(F, [b**3, a**2])
assert is_isomorphic(G, H)
# Trivial Case
# FpGroup -> FpGroup
H = FpGroup(F, [a**3, b**3, (a*b)**2])
F, c, d = free_group("c, d")
G = FpGroup(F, [c**3, d**3, (c*d)**2])
check, T = group_isomorphism(G, H)
assert check
T(c**3*d**2) == a**3*b**2
# FpGroup -> PermutationGroup
# FpGroup is converted to the equivalent isomorphic group.
F, a, b = free_group("a, b")
G = FpGroup(F, [a**3, b**3, (a*b)**2])
H = AlternatingGroup(4)
check, T = group_isomorphism(G, H)
assert check
assert T(b*a*b**-1*a**-1*b**-1) == Permutation(0, 2, 3)
assert T(b*a*b*a**-1*b**-1) == Permutation(0, 3, 2)
# PermutationGroup -> PermutationGroup
D = DihedralGroup(8)
p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
P = PermutationGroup(p)
assert not is_isomorphic(D, P)
A = CyclicGroup(5)
B = CyclicGroup(7)
assert not is_isomorphic(A, B)
# Two groups of the same prime order are isomorphic to each other.
G = FpGroup(F, [a, b**5])
H = CyclicGroup(5)
assert G.order() == H.order()
assert is_isomorphic(G, H)
def test_homomorphism():
# FpGroup -> PermutationGroup
F, a, b = free_group("a, b")
G = FpGroup(F, [a**3, b**3, (a*b)**2])
c = Permutation(3)(0, 1, 2)
d = Permutation(3)(1, 2, 3)
A = AlternatingGroup(4)
T = homomorphism(G, A, [a, b], [c, d])
assert T(a*b**2*a**-1) == c*d**2*c**-1
assert T.is_isomorphism()
assert T(T.invert(Permutation(3)(0, 2, 3))) == Permutation(3)(0, 2, 3)
T = homomorphism(G, AlternatingGroup(4), G.generators)
assert T.is_trivial()
assert T.kernel().order() == G.order()
E, e = free_group("e")
G = FpGroup(E, [e**8])
P = PermutationGroup([Permutation(0, 1, 2, 3), Permutation(0, 2)])
T = homomorphism(G, P, [e], [Permutation(0, 1, 2, 3)])
assert T.image().order() == 4
assert T(T.invert(Permutation(0, 2)(1, 3))) == Permutation(0, 2)(1, 3)
T = homomorphism(E, AlternatingGroup(4), E.generators, [c])
assert T.invert(c**2) == e**-1 #order(c) == 3 so c**2 == c**-1
# FreeGroup -> FreeGroup
T = homomorphism(F, E, [a], [e])
assert T(a**-2*b**4*a**2).is_identity
# FreeGroup -> FpGroup
G = FpGroup(F, [a*b*a**-1*b**-1])
T = homomorphism(F, G, F.generators, G.generators)
assert T.invert(a**-1*b**-1*a**2) == a*b**-1
# PermutationGroup -> PermutationGroup
D = DihedralGroup(8)
p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
P = PermutationGroup(p)
T = homomorphism(P, D, [p], [p])
assert T.is_injective()
assert not T.is_isomorphism()
assert T.invert(p**3) == p**3
T2 = homomorphism(F, P, [F.generators[0]], P.generators)
T = T.compose(T2)
assert T.domain == F
assert T.codomain == D
assert T(a*b) == p
def test_homomorphism():
# FpGroup -> PermutationGroup
F, a, b = free_group("a, b")
G = FpGroup(F, [a**3, b**3, (a*b)**2])
c = Permutation(3)(0, 1, 2)
d = Permutation(3)(1, 2, 3)
A = AlternatingGroup(4)
T = homomorphism(G, A, [a, b], [c, d])
assert T(a*b**2*a**-1) == c*d**2*c**-1
assert T.is_isomorphism()
assert T(T.invert(Permutation(3)(0, 2, 3))) == Permutation(3)(0, 2, 3)
T = homomorphism(G, AlternatingGroup(4), G.generators)
assert T.is_trivial()
assert T.kernel().order() == G.order()
E, e = free_group("e")
G = FpGroup(E, [e**8])
P = PermutationGroup([Permutation(0, 1, 2, 3), Permutation(0, 2)])
T = homomorphism(G, P, [e], [Permutation(0, 1, 2, 3)])
assert T.image().order() == 4
assert T(T.invert(Permutation(0, 2)(1, 3))) == Permutation(0, 2)(1, 3)
T = homomorphism(E, AlternatingGroup(4), E.generators, [c])
assert T.invert(c**2) == e**-1 #order(c) == 3 so c**2 == c**-1
# FreeGroup -> FreeGroup
T = homomorphism(F, E, [a], [e])
assert T(a**-2*b**4*a**2).is_identity
# FreeGroup -> FpGroup
G = FpGroup(F, [a*b*a**-1*b**-1])
T = homomorphism(F, G, F.generators, G.generators)
assert T.invert(a**-1*b**-1*a**2) == a*b**-1
# PermutationGroup -> PermutationGroup
D = DihedralGroup(8)
p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
P = PermutationGroup(p)
T = homomorphism(P, D, [p], [p])
assert T.is_injective()
assert not T.is_isomorphism()
assert T.invert(p**3) == p**3
T2 = homomorphism(F, P, [F.generators[0]], P.generators)
T = T.compose(T2)
assert T.domain == F
assert T.codomain == D
assert T(a*b) == p
