def test_fp_subgroup():
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x*y*x**-1*y])
S = FpSubgroup(f, [x*y])
assert (x*y)**-3 in S
K, T = f.subgroup([x*y], homomorphism=True)
assert T(K.generators) == [y*x**-1]
_test_subgroup(K, T, S)
S = FpSubgroup(f, [x**-1*y*x])
assert x**-1*y**4*x in S
assert x**-1*y**4*x**2 not in S
K, T = f.subgroup([x**-1*y*x], homomorphism=True)
assert T(K.generators[0]**3) == y**3
_test_subgroup(K, T, S)
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
K, T = f.subgroup(H, homomorphism=True)
S = FpSubgroup(f, H)
_test_subgroup(K, T, S)
def test_fp_subgroup():
def _test_subgroup(K, T, S):
_gens = T(K.generators)
assert all(elem in S for elem in _gens)
assert T.is_injective()
assert T.image().order() == S.order()
F, x, y = free_group("x, y")
f = FpGroup(F, [x**4, y**2, x * y * x**-1 * y])
S = FpSubgroup(f, [x * y])
assert (x * y)**-3 in S
K, T = f.subgroup([x * y], homomorphism=True)
assert T(K.generators) == [y * x**-1]
_test_subgroup(K, T, S)
S = FpSubgroup(f, [x**-1 * y * x])
assert x**-1 * y**4 * x in S
assert x**-1 * y**4 * x**2 not in S
K, T = f.subgroup([x**-1 * y * x], homomorphism=True)
assert T(K.generators[0]**3) == y**3
_test_subgroup(K, T, S)
f = FpGroup(F, [x**3, y**5, (x * y)**2])
H = [x * y, x**-1 * y**-1 * x * y * x]
K, T = f.subgroup(H, homomorphism=True)
S = FpSubgroup(f, H)
_test_subgroup(K, T, S)
def test_subgroup_presentations():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
p1 = reidemeister_presentation(f, H)
assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"
H = f.subgroup(H)
assert (H.generators, H.relators) == p1
f = FpGroup(F, [x**3, y**3, (x*y)**3])
H = [x*y, x*y**-1]
p2 = reidemeister_presentation(f, H)
assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"
f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
H = [x]
p3 = reidemeister_presentation(f, H)
assert str(p3) == "((x_0,), (x_0**4,))"
f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
H = [x]
p4 = reidemeister_presentation(f, H)
assert str(p4) == "((x_0,), (x_0**6,))"
# this presentation can be improved, the most simplified form
# of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
# See [2] Pg 474 group PSL_2(11)
# This is the group PSL_2(11)
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [a, b, c**2]
gens, rels = reidemeister_presentation(f, H)
assert str(gens) == "(b_1, c_3)"
assert len(rels) == 18
def test_subgroup_presentations():
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**5, (x*y)**2])
H = [x*y, x**-1*y**-1*x*y*x]
p1 = reidemeister_presentation(f, H)
assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"
H = f.subgroup(H)
assert (H.generators, H.relators) == p1
f = FpGroup(F, [x**3, y**3, (x*y)**3])
H = [x*y, x*y**-1]
p2 = reidemeister_presentation(f, H)
assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"
f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
H = [x]
p3 = reidemeister_presentation(f, H)
assert str(p3) == "((x_0,), (x_0**4,))"
f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
H = [x]
p4 = reidemeister_presentation(f, H)
assert str(p4) == "((x_0,), (x_0**6,))"
# this presentation can be improved, the most simplified form
# of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
# See [2] Pg 474 group PSL_2(11)
# This is the group PSL_2(11)
F, a, b, c = free_group("a, b, c")
f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
H = [a, b, c**2]
gens, rels = reidemeister_presentation(f, H)
assert str(gens) == "(b_1, c_3)"
assert len(rels) == 18
