def test_permutation_methods():
from sympy.combinatorics.fp_groups import FpSubgroup
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x*y*x**-1*y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x*y*x**-1*y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1*x*y in S
# Z_5xZ_4
G = FpGroup(F, [x*y*x**-1*y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x*y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x*y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
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_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_cyclic():
F, x, y = free_group("x, y")
f = FpGroup(F, [x * y, x**-1 * y**-1 * x * y * x])
assert f.is_cyclic
f = FpGroup(F, [x * y, x * y**-1])
assert f.is_cyclic
f = FpGroup(F, [x**4, y**2, x * y * x**-1 * y])
assert not f.is_cyclic
def test_abelian_invariants():
F, x, y = free_group("x, y")
f = FpGroup(F, [x * y, x**-1 * y**-1 * x * y * x])
assert f.abelian_invariants() == []
f = FpGroup(F, [x * y, x * y**-1])
assert f.abelian_invariants() == [2]
f = FpGroup(F, [x**4, y**2, x * y * x**-1 * y])
assert f.abelian_invariants() == [2, 4]
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_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_look_ahead():
# Section 3.2 [Test Example] Example (d) from [2]
F, a, b, c = free_group("a, b, c")
f = FpGroup(
F,
[
a ** 11,
b ** 5,
c ** 4,
(a * c) ** 3,
b ** 2 * c ** -1 * b ** -1 * c,
a ** 4 * b ** -1 * a ** -1 * b,
],
)
H = [c, b, c ** 2]
table0 = [
[1, 2, 0, 0, 0, 0],
[3, 0, 4, 5, 6, 7],
[0, 8, 9, 10, 11, 12],
[5, 1, 10, 13, 14, 15],
[16, 5, 16, 1, 17, 18],
[4, 3, 1, 8, 19, 20],
[12, 21, 22, 23, 24, 1],
[25, 26, 27, 28, 1, 24],
[2, 10, 5, 16, 22, 28],
[10, 13, 13, 2, 29, 30],
]
CosetTable.max_stack_size = 10
C_c = coset_enumeration_c(f, H)
C_c.compress()
C_c.standardize()
assert C_c.table[:10] == table0
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))"
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]
k = ("((a_0, b_1, c_3), "
"(c_3**3, b_1**5, a_0**4*b_1**-2*a_0**-1*b_1**2, "
"c_3*b_1**-2*c_3*b_1**-2*c_3*b_1**-2, b_1**-1*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**2*c_3**-1, "
"a_0**11, a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*c_3, "
"a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
"a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
"a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
"b_1**-2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2, "
"b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
"b_1**2*c_3*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*c_3, "
"b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**-2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
"c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
"c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2, "
"b_1**2*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1**2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
"c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0, "
"a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1))"
)
assert str(reidemeister_presentation(f, H)) == k
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a*b*a**-1*b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent == False
R.make_confluent()
assert R.is_confluent == True
assert G.reduce(b**3*a**4*b**-2*a) == a**5*b
assert G.equals(b**2*a**-1*b, b**4*a**-1*b**-1)
G = FpGroup(F, [a**3, b**3, (a*b)**2])
R = G._rewriting_system
R.make_confluent()
assert R.is_confluent
assert G.reduce(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
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_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_fp_subgroup():
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
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
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_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_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_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_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a*b*a**-1*b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent
assert G.reduce(b**-1*a) == a*b**-1
assert G.reduce(b**3*a**4*b**-2*a) == a**5*b
assert G.equals(b**2*a**-1*b, b**4*a**-1*b**-1)
assert R.reduce_using_automaton(b*a*a**2*b**-1) == a**3
assert R.reduce_using_automaton(b**3*a**4*b**-2*a) == a**5*b
assert R.reduce_using_automaton(b**-1*a) == a*b**-1
G = FpGroup(F, [a**3, b**3, (a*b)**2])
R = G._rewriting_system
R.make_confluent()
# R._is_confluent should be set to True after
# a successful run of make_confluent
assert R.is_confluent
# but also the system should actually be confluent
assert R._check_confluence()
assert G.reduce(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
# check for automaton reduction
assert R.reduce_using_automaton(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
G = FpGroup(F, [a**2, b**3, (a*b)**4])
R = G._rewriting_system
assert G.reduce(a**2*b**-2*a**2*b) == b**-1
assert R.reduce_using_automaton(a**2*b**-2*a**2*b) == b**-1
assert G.reduce(a**3*b**-2*a**2*b) == a**-1*b**-1
assert R.reduce_using_automaton(a**3*b**-2*a**2*b) == a**-1*b**-1
# Check after adding a rule
R.add_rule(a**2, b)
assert R.reduce_using_automaton(a**2*b**-2*a**2*b) == b**-1
assert R.reduce_using_automaton(a**4*b**-2*a**2*b**3) == b
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a * b * a**-1 * b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent
assert G.reduce(b**-1 * a) == a * b**-1
assert G.reduce(b**3 * a**4 * b**-2 * a) == a**5 * b
assert G.equals(b**2 * a**-1 * b, b**4 * a**-1 * b**-1)
G = FpGroup(F, [a**3, b**3, (a * b)**2])
R = G._rewriting_system
R.make_confluent()
# R._is_confluent should be set to True after
# a successful run of make_confluent
assert R.is_confluent
# but also the system should actually be confluent
assert R._check_confluence()
assert G.reduce(b * a**-1 * b**-1 * a**3 * b**4 * a**-1 *
b**-15) == a**-1 * b**-1
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a * b * a**-1 * b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent == False
R.make_confluent()
assert R.is_confluent == True
assert G.reduce(b**3 * a**4 * b**-2 * a) == a**5 * b
assert G.equals(b**2 * a**-1 * b, b**4 * a**-1 * b**-1)
G = FpGroup(F, [a**3, b**3, (a * b)**2])
R = G._rewriting_system
R.make_confluent()
assert R.is_confluent
assert G.reduce(b * a**-1 * b**-1 * a**3 * b**4 * a**-1 *
b**-15) == a**-1 * b**-1
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a*b*a**-1*b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent
assert G.reduce(b**-1*a) == a*b**-1
assert G.reduce(b**3*a**4*b**-2*a) == a**5*b
assert G.equals(b**2*a**-1*b, b**4*a**-1*b**-1)
G = FpGroup(F, [a**3, b**3, (a*b)**2])
R = G._rewriting_system
R.make_confluent()
# R._is_confluent should be set to True after
# a successful run of make_confluent
assert R.is_confluent
# but also the system should actually be confluent
assert R._check_confluence()
assert G.reduce(b*a**-1*b**-1*a**3*b**4*a**-1*b**-15) == a**-1*b**-1
def test_rewriting():
F, a, b = free_group("a, b")
G = FpGroup(F, [a * b * a**-1 * b**-1])
a, b = G.generators
R = G._rewriting_system
assert R.is_confluent
assert G.reduce(b**-1 * a) == a * b**-1
assert G.reduce(b**3 * a**4 * b**-2 * a) == a**5 * b
assert G.equals(b**2 * a**-1 * b, b**4 * a**-1 * b**-1)
assert R.reduce_using_automaton(b * a * a**2 * b**-1) == a**3
assert R.reduce_using_automaton(b**3 * a**4 * b**-2 * a) == a**5 * b
assert R.reduce_using_automaton(b**-1 * a) == a * b**-1
G = FpGroup(F, [a**3, b**3, (a * b)**2])
R = G._rewriting_system
R.make_confluent()
# R._is_confluent should be set to True after
# a successful run of make_confluent
assert R.is_confluent
# but also the system should actually be confluent
assert R._check_confluence()
assert G.reduce(b * a**-1 * b**-1 * a**3 * b**4 * a**-1 *
b**-15) == a**-1 * b**-1
# check for automaton reduction
assert R.reduce_using_automaton(b * a**-1 * b**-1 * a**3 * b**4 * a**-1 *
b**-15) == a**-1 * b**-1
G = FpGroup(F, [a**2, b**3, (a * b)**4])
R = G._rewriting_system
assert G.reduce(a**2 * b**-2 * a**2 * b) == b**-1
assert R.reduce_using_automaton(a**2 * b**-2 * a**2 * b) == b**-1
assert G.reduce(a**3 * b**-2 * a**2 * b) == a**-1 * b**-1
assert R.reduce_using_automaton(a**3 * b**-2 * a**2 * b) == a**-1 * b**-1
# Check after adding a rule
R.add_rule(a**2, b)
assert R.reduce_using_automaton(a**2 * b**-2 * a**2 * b) == b**-1
assert R.reduce_using_automaton(a**4 * b**-2 * a**2 * b**3) == b
def test_low_index_subgroups():
F, x, y = free_group("x, y")
# Example 5.10 from [1] Pg. 194
f = FpGroup(F, [x**2, y**3, (x*y)**4])
L = low_index_subgroups(f, 4)
t1 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]],
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]],
[[1, 1, 0, 0], [0, 0, 1, 1]]]
for i in range(len(t1)):
assert L[i].table == t1[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 15)
t2 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[4, 4, 5, 3], [6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [9, 9, 6, 8], [12, 12, 11, 7], [8, 8, 7, 10],
[10, 10, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [10, 10, 11, 7], [8, 8, 7, 10],
[9, 9, 13, 14], [14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5],
[6, 6, 5, 3], [7, 7, 3, 4], [4, 4, 8, 9], [5, 5, 10, 11],
[11, 11, 9, 6], [12, 12, 6, 8], [13, 13, 11, 7], [8, 8, 7, 10],
[9, 9, 12, 12], [10, 10, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [11, 11, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 3, 3], [2, 2, 5, 6]
, [7, 7, 6, 4], [8, 8, 4, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 4, 4]
, [7, 7, 6, 3], [8, 8, 3, 5], [5, 5, 8, 9], [6, 6, 9, 7],
[10, 10, 7, 8], [9, 9, 11, 12], [13, 13, 12, 10], [12, 12, 10, 11],
[11, 11, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [5, 5, 6, 3], [9, 9, 3, 5], [10, 10, 8, 4], [8, 8, 4, 7],
[6, 6, 10, 11], [7, 7, 11, 9], [12, 12, 9, 10], [11, 11, 13, 14],
[14, 14, 14, 12], [13, 13, 12, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 10, 12], [7, 7, 12, 9], [8, 8, 11, 11], [13, 13, 9, 10],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [6, 6, 3, 5], [10, 10, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [7, 7, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 9, 9], [6, 6, 11, 12], [8, 8, 12, 10], [13, 13, 10, 11],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [7, 7, 8, 4], [11, 11, 4, 7],
[5, 5, 12, 11], [6, 6, 10, 10], [8, 8, 9, 12], [13, 13, 11, 9],
[12, 12, 13, 13]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8]
, [9, 9, 6, 3], [10, 10, 3, 5], [11, 11, 8, 4], [12, 12, 4, 7],
[5, 5, 9, 9], [6, 6, 12, 13], [7, 7, 11, 11], [8, 8, 13, 10],
[13, 13, 10, 12]],
[[1, 1, 0, 0], [0, 0, 2, 3], [4, 4, 3, 1], [5, 5, 1, 2], [2, 2, 4, 4]
, [3, 3, 6, 7], [7, 7, 7, 5], [6, 6, 5, 6]]]
for i in range(len(t2)):
assert L[i].table == t2[i]
f = FpGroup(F, [x**2, y**3, (x*y)**7])
L = low_index_subgroups(f, 10, [x])
t3 = [[[0, 0, 0, 0]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [4, 4, 5, 3],
[6, 6, 3, 4], [5, 5, 6, 6]],
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 4, 5], [6, 6, 5, 3],
[5, 5, 3, 4], [4, 4, 6, 6]],
[[0, 0, 1, 2], [3, 3, 2, 0], [4, 4, 0, 1], [1, 1, 5, 6], [2, 2, 7, 8],
[6, 6, 6, 3], [5, 5, 3, 5], [8, 8, 8, 4], [7, 7, 4, 7]]]
for i in range(len(t3)):
assert L[i].table == t3[i]
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_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_modified_methods():
"""
Tests for modified coset table methods.
Example 5.7 from [1] Holt, D., Eick, B., O'Brien
"Handbook of Computational Group Theory".
"""
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]
C = CosetTable(f, H)
C.modified_define(0, x)
identity = C._grp.identity
a_0 = C._grp.generators[0]
a_1 = C._grp.generators[1]
assert C.P == [[identity, None, None, None], [None, identity, None, None]]
assert C.table == [[1, None, None, None], [None, 0, None, None]]
C.modified_define(1, x)
assert C.table == [[1, None, None, None], [2, 0, None, None], [None, 1, None, None]]
assert C.P == [
[identity, None, None, None],
[identity, identity, None, None],
[None, identity, None, None],
]
C.modified_scan(0, x ** 3, C._grp.identity, fill=False)
assert C.P == [
[identity, identity, None, None],
[identity, identity, None, None],
[identity, identity, None, None],
]
assert C.table == [[1, 2, None, None], [2, 0, None, None], [0, 1, None, None]]
C.modified_scan(0, x * y, C._grp.generators[0], fill=False)
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, None],
]
assert C.table == [[1, 2, None, 1], [2, 0, 0, None], [0, 1, None, None]]
C.modified_define(2, y ** -1)
assert C.table == [
[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[None, None, 2, None],
]
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[None, None, identity, None],
]
C.modified_scan(0, x ** -1 * y ** -1 * x * y * x, C._grp.generators[1])
assert C.table == [
[1, 2, None, 1],
[2, 0, 0, None],
[0, 1, None, 3],
[3, 3, 2, None],
]
assert C.P == [
[identity, identity, None, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1 ** -1, identity, None],
]
C.modified_scan(2, (x * y) ** 2, C._grp.identity)
assert C.table == [[1, 2, 3, 1], [2, 0, 0, None], [0, 1, None, 3], [3, 3, 2, 0]]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, None, identity],
[a_1, a_1 ** -1, identity, a_1],
]
C.modified_define(2, y)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, None],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, None, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, None],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[None, None, None, identity],
]
C.modified_scan(0, y ** 5, C._grp.identity)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[None, None, 1, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, a_0 * a_1 ** -1],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[None, None, a_1 * a_0 ** -1, identity],
]
C.modified_scan(1, (x * y) ** 2, C._grp.identity)
assert C.table == [
[1, 2, 3, 1],
[2, 0, 0, 4],
[0, 1, 4, 3],
[3, 3, 2, 0],
[4, 4, 1, 2],
]
assert C.P == [
[identity, identity, a_1 ** -1, a_0 ** -1],
[identity, identity, a_0, a_0 * a_1 ** -1],
[identity, identity, identity, identity],
[a_1, a_1 ** -1, identity, a_1],
[a_0 * a_1 ** -1, a_1 * a_0 ** -1, a_1 * a_0 ** -1, identity],
]
# Modified coset enumeration test
f = FpGroup(F, [x ** 3, y ** 3, x ** -1 * y ** -1 * x * y])
C = coset_enumeration_r(f, [x])
C_m = modified_coset_enumeration_r(f, [x])
assert C_m.table == C.table
def test_coset_enumeration():
# this test function contains the combined tests for the two strategies
# i.e. HLT and Felsch strategies.
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**3, x**-1 * y**-1 * x * y])
C_r = coset_enumeration_r(f, [x])
C_r.compress()
C_r.standardize()
C_c = coset_enumeration_c(f, [x])
C_c.compress()
C_c.standardize()
table1 = [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]
assert C_r.table == table1
assert C_c.table == table1
# E₁ from [2] Pg. 474
F, r, s, t = free_group("r, s, t")
E1 = FpGroup(
F,
[t**-1 * r * t * r**-2, r**-1 * s * r * s**-2, s**-1 * t * s * t**-2])
C_r = coset_enumeration_r(E1, [])
C_r.compress()
C_c = coset_enumeration_c(E1, [])
C_c.compress()
table2 = [[0, 0, 0, 0, 0, 0]]
assert C_r.table == table2
# test for issue #11449
assert C_c.table == table2
# Cox group from [2] Pg. 474
F, a, b = free_group("a, b")
Cox = FpGroup(F,
[a**6, b**6, (a * b)**2, (a**2 * b**2)**2, (a**3 * b**3)**5])
C_r = coset_enumeration_r(Cox, [a])
C_r.compress()
C_r.standardize()
C_c = coset_enumeration_c(Cox, [a])
C_c.compress()
C_c.standardize()
table3 = [[0, 0, 1, 2], [2, 3, 4, 0], [5, 1, 0, 6], [1, 7, 8, 9],
[9, 10, 11, 1], [12, 2, 9, 13], [14, 9, 2, 11], [3, 12, 15, 16],
[16, 17, 18, 3], [6, 4, 3, 5], [4, 19, 20, 21], [21, 22, 6, 4],
[7, 5, 23, 24], [25, 23, 5, 18], [19, 6, 22,
26], [24, 27, 28, 7],
[29, 8, 7, 30], [8, 31, 32, 33], [33, 34, 13, 8],
[10, 14, 35, 35], [35, 36, 37, 10], [30, 11, 10, 29],
[11, 38, 39, 14], [13, 39, 38, 12], [40, 15, 12, 41],
[42, 13, 34, 43], [44, 35, 14, 45], [15, 46, 47, 34],
[34, 48, 49, 15], [50, 16, 21, 51], [52, 21, 16, 49],
[17, 50, 53, 54], [54, 55, 56, 17], [41, 18, 17, 40],
[18, 28, 27, 25], [26, 20, 19, 19], [20, 57, 58, 59],
[59, 60, 51, 20], [22, 52, 61, 23], [23, 62, 63, 22],
[64, 24, 33, 65], [48, 33, 24, 61], [62, 25, 54, 66],
[67, 54, 25, 68], [57, 26, 59, 69], [70, 59, 26, 63],
[27, 64, 71, 72], [72, 73, 68, 27], [28, 41, 74, 75],
[75, 76, 30, 28], [31, 29, 77, 78], [79, 77, 29, 37],
[38, 30, 76, 80], [78, 81, 82, 31], [43, 32, 31, 42],
[32, 83, 84, 85], [85, 86, 65, 32], [36, 44, 87, 88],
[88, 89, 90, 36], [45, 37, 36, 44], [37, 82, 81, 79],
[80, 74, 41, 38], [39, 42, 91, 92], [92, 93, 45, 39],
[46, 40, 94, 95], [96, 94, 40, 56], [97, 91, 42, 82],
[83, 43, 98, 99], [100, 98, 43, 47], [101, 87, 44, 90],
[82, 45, 93, 97], [95, 102, 103, 46], [104, 47, 46, 105],
[47, 106, 107, 100], [61, 108, 109, 48], [105, 49, 48, 104],
[49, 110, 111, 52], [51, 111, 110, 50], [112, 53, 50, 113],
[114, 51, 60, 115], [116, 61, 52, 117], [53, 118, 119, 60],
[60, 70, 66, 53], [55, 67, 120, 121], [121, 122, 123, 55],
[113, 56, 55, 112], [56, 103, 102, 96], [69, 124, 125, 57],
[115, 58, 57, 114], [58, 126, 127, 128], [128, 128, 69, 58],
[66, 129, 130, 62], [117, 63, 62, 116], [63, 125, 124, 70],
[65, 109, 108, 64], [131, 71, 64, 132], [133, 65, 86, 134],
[135, 66, 70, 136], [68, 130, 129, 67], [137, 120, 67, 138],
[132, 68, 73, 131], [139, 69, 128, 140], [71, 141, 142, 86],
[86, 143, 144, 71], [145, 72, 75, 146], [147, 75, 72, 144],
[73, 145, 148, 120], [120, 149, 150, 73], [74, 151, 152, 94],
[94, 153, 146, 74], [76, 147, 154, 77], [77, 155, 156, 76],
[157, 78, 85, 158], [143, 85, 78, 154], [155, 79, 88, 159],
[160, 88, 79, 161], [151, 80, 92, 162], [163, 92, 80, 156],
[81, 157, 164, 165], [165, 166, 161, 81], [99, 107, 106, 83],
[134, 84, 83, 133], [84, 167, 168, 169], [169, 170, 158, 84],
[87, 171, 172, 93], [93, 163, 159, 87], [89, 160, 173, 174],
[174, 175, 176, 89], [90, 90, 89, 101], [91, 177, 178, 98],
[98, 179, 162, 91], [180, 95, 100, 181], [179, 100, 95, 152],
[153, 96, 121, 148], [182, 121, 96, 183], [177, 97, 165, 184],
[185, 165, 97, 172], [186, 99, 169, 187], [188, 169, 99, 178],
[171, 101, 174, 189], [190, 174, 101, 176], [102, 180, 191, 192],
[192, 193, 183, 102], [103, 113, 194, 195], [195, 196, 105, 103],
[106, 104, 197, 198], [199, 197, 104, 109], [110, 105, 196, 200],
[198, 201, 133, 106], [107, 186, 202, 203], [203, 204, 181, 107],
[108, 116, 205, 206], [206, 207, 132, 108], [109, 133, 201, 199],
[200, 194, 113, 110], [111, 114, 208, 209], [209, 210, 117, 111],
[118, 112, 211, 212], [213, 211, 112, 123], [214, 208, 114, 125],
[126, 115, 215, 216], [217, 215, 115, 119], [218, 205, 116, 130],
[125, 117, 210, 214], [212, 219, 220, 118], [136, 119, 118, 135],
[119, 221, 222, 217], [122, 182, 223, 224], [224, 225, 226, 122],
[138, 123, 122, 137], [123, 220, 219, 213], [124, 139, 227, 228],
[228, 229, 136, 124], [216, 222, 221, 126], [140, 127, 126, 139],
[127, 230, 231, 232], [232, 233, 140, 127], [129, 135, 234, 235],
[235, 236, 138, 129], [130, 132, 207, 218], [141, 131, 237, 238],
[239, 237, 131, 150], [167, 134, 240, 241], [242, 240, 134, 142],
[243, 234, 135, 220], [221, 136, 229, 244], [149, 137, 245, 246],
[247, 245, 137, 226], [220, 138, 236, 243], [244, 227, 139, 221],
[230, 140, 233, 248], [238, 249, 250, 141], [251, 142, 141, 252],
[142, 253, 254, 242], [154, 255, 256, 143], [252, 144, 143, 251],
[144, 257, 258, 147], [146, 258, 257, 145], [259, 148, 145, 260],
[261, 146, 153, 262], [263, 154, 147, 264], [148, 265, 266, 153],
[246, 267, 268, 149], [260, 150, 149, 259], [150, 250, 249, 239],
[162, 269, 270, 151], [262, 152, 151, 261], [152, 271, 272, 179],
[159, 273, 274, 155], [264, 156, 155, 263], [156, 270, 269, 163],
[158, 256, 255, 157], [275, 164, 157, 276], [277, 158, 170, 278],
[279, 159, 163, 280], [161, 274, 273, 160], [281, 173, 160, 282],
[276, 161, 166, 275], [283, 162, 179, 284], [164, 285, 286, 170],
[170, 188, 184, 164], [166, 185, 189, 173], [173, 287, 288, 166],
[241, 254, 253, 167], [278, 168, 167, 277], [168, 289, 290, 291],
[291, 292, 187, 168], [189, 293, 294, 171], [280, 172, 171, 279],
[172, 295, 296, 185], [175, 190, 297, 297], [297, 298, 299, 175],
[282, 176, 175, 281], [176, 294, 293, 190], [184, 296, 295, 177],
[284, 178, 177, 283], [178, 300, 301, 188], [181, 272, 271, 180],
[302, 191, 180, 303], [304, 181, 204, 305], [183, 266, 265, 182],
[306, 223, 182, 307], [303, 183, 193, 302], [308, 184, 188, 309],
[310, 189, 185, 311], [187, 301, 300, 186], [305, 202, 186, 304],
[312, 187, 292, 313], [314, 297, 190, 315], [191, 316, 317, 204],
[204, 318, 319, 191], [320, 192, 195, 321], [322, 195, 192, 319],
[193, 320, 323, 223], [223, 324, 325, 193], [194, 326, 327, 211],
[211, 328, 321, 194], [196, 322, 329, 197], [197, 330, 331, 196],
[332, 198, 203, 333], [318, 203, 198, 329], [330, 199, 206, 334],
[335, 206, 199, 336], [326, 200, 209, 337], [338, 209, 200, 331],
[201, 332, 339, 240], [240, 340, 336, 201], [202, 341, 342, 292],
[292, 343, 333, 202], [205, 344, 345, 210], [210, 338, 334, 205],
[207, 335, 346, 237], [237, 347, 348, 207], [208, 349, 350, 215],
[215, 351, 337, 208], [352, 212, 217, 353], [351, 217, 212, 327],
[328, 213, 224, 323], [354, 224, 213, 355], [349, 214, 228, 356],
[357, 228, 214, 345], [358, 216, 232, 359], [360, 232, 216, 350],
[344, 218, 235, 361], [362, 235, 218, 348], [219, 352, 363, 364],
[364, 365, 355, 219], [222, 358, 366, 367], [367, 368, 353, 222],
[225, 354, 369, 370], [370, 371, 372, 225], [307, 226, 225, 306],
[226, 268, 267, 247], [227, 373, 374, 233], [233, 360, 356, 227],
[229, 357, 361, 234], [234, 375, 376, 229], [248, 231, 230, 230],
[231, 377, 378, 379], [379, 380, 359, 231], [236, 362, 381, 245],
[245, 382, 383, 236], [384, 238, 242, 385], [340, 242, 238, 346],
[347, 239, 246, 381], [386, 246, 239, 387], [388, 241, 291, 389],
[343, 291, 241, 339], [375, 243, 364, 390], [391, 364, 243, 383],
[373, 244, 367, 392], [393, 367, 244, 376], [382, 247, 370, 394],
[395, 370, 247, 396], [377, 248, 379, 397], [398, 379, 248, 374],
[249, 384, 399, 400], [400, 401, 387, 249], [250, 260, 402, 403],
[403, 404, 252, 250], [253, 251, 405, 406], [407, 405, 251, 256],
[257, 252, 404, 408], [406, 409, 277, 253], [254, 388, 410, 411],
[411, 412, 385, 254], [255, 263, 413, 414], [414, 415, 276, 255],
[256, 277, 409, 407], [408, 402, 260, 257], [258, 261, 416, 417],
[417, 418, 264, 258], [265, 259, 419, 420], [421, 419, 259, 268],
[422, 416, 261, 270], [271, 262, 423, 424], [425, 423, 262, 266],
[426, 413, 263, 274], [270, 264, 418, 422], [420, 427, 307, 265],
[266, 303, 428, 425], [267, 386, 429, 430], [430, 431, 396, 267],
[268, 307, 427, 421], [269, 283, 432, 433], [433, 434, 280, 269],
[424, 428, 303, 271], [272, 304, 435, 436], [436, 437, 284, 272],
[273, 279, 438, 439], [439, 440, 282, 273], [274, 276, 415, 426],
[285, 275, 441, 442], [443, 441, 275, 288], [289, 278, 444, 445],
[446, 444, 278, 286], [447, 438, 279, 294], [295, 280, 434, 448],
[287, 281, 449, 450], [451, 449, 281, 299], [294, 282, 440, 447],
[448, 432, 283, 295], [300, 284, 437, 452], [442, 453, 454, 285],
[309, 286, 285, 308], [286, 455, 456, 446], [450, 457, 458, 287],
[311, 288, 287, 310], [288, 454, 453, 443], [445, 456, 455, 289],
[313, 290, 289, 312], [290, 459, 460, 461], [461, 462, 389, 290],
[293, 310, 463, 464], [464, 465, 315, 293], [296, 308, 466, 467],
[467, 468, 311, 296], [298, 314, 469, 470], [470, 471, 472, 298],
[315, 299, 298, 314], [299, 458, 457, 451], [452, 435, 304, 300],
[301, 312, 473, 474], [474, 475, 309, 301], [316, 302, 476, 477],
[478, 476, 302, 325], [341, 305, 479, 480], [481, 479, 305, 317],
[324, 306, 482, 483], [484, 482, 306, 372], [485, 466, 308, 454],
[455, 309, 475, 486], [487, 463, 310, 458], [454, 311, 468, 485],
[486, 473, 312, 455], [459, 313, 488, 489], [490, 488, 313, 342],
[491, 469, 314, 472], [458, 315, 465, 487], [477, 492, 485, 316],
[463, 317, 316, 468], [317, 487, 493, 481], [329, 447, 464, 318],
[468, 319, 318, 463], [319, 467, 448, 322], [321, 448, 467, 320],
[475, 323, 320, 466], [432, 321, 328, 437], [438, 329, 322, 434],
[323, 474, 452, 328], [483, 494, 486, 324], [466, 325, 324, 475],
[325, 485, 492, 478], [337, 422, 433, 326], [437, 327, 326, 432],
[327, 436, 424, 351], [334, 426, 439, 330], [434, 331, 330, 438],
[331, 433, 422, 338], [333, 464, 447, 332], [449, 339, 332, 440],
[465, 333, 343, 469], [413, 334, 338, 418], [336, 439, 426, 335],
[441, 346, 335, 415], [440, 336, 340, 449], [416, 337, 351, 423],
[339, 451, 470, 343], [346, 443, 450, 340], [480, 493, 487, 341],
[469, 342, 341, 465], [342, 491, 495, 490], [361, 407, 414, 344],
[418, 345, 344, 413], [345, 417, 408, 357], [381, 446, 442, 347],
[415, 348, 347, 441], [348, 414, 407, 362], [356, 408, 417, 349],
[423, 350, 349, 416], [350, 425, 420, 360], [353, 424, 436, 352],
[479, 363, 352, 435], [428, 353, 368, 476], [355, 452, 474, 354],
[488, 369, 354, 473], [435, 355, 365, 479], [402, 356, 360, 419],
[405, 361, 357, 404], [359, 420, 425, 358], [476, 366, 358, 428],
[427, 359, 380, 482], [444, 381, 362, 409], [363, 481, 477, 368],
[368, 393, 390, 363], [365, 391, 394, 369], [369, 490, 480, 365],
[366, 478, 483, 380], [380, 398, 392, 366], [371, 395, 496, 497],
[497, 498, 489, 371], [473, 372, 371, 488], [372, 486, 494, 484],
[392, 400, 403, 373], [419, 374, 373, 402], [374, 421, 430, 398],
[390, 411, 406, 375], [404, 376, 375, 405], [376, 403, 400, 393],
[397, 430, 421, 377], [482, 378, 377, 427], [378, 484, 497, 499],
[499, 499, 397, 378], [394, 461, 445, 382], [409, 383, 382, 444],
[383, 406, 411, 391], [385, 450, 443, 384], [492, 399, 384, 453],
[457, 385, 412, 493], [387, 442, 446, 386], [494, 429, 386, 456],
[453, 387, 401, 492], [389, 470, 451, 388], [493, 410, 388, 457],
[471, 389, 462, 495], [412, 390, 393, 399], [462, 394, 391, 410],
[401, 392, 398, 429], [396, 445, 461, 395], [498, 496, 395, 460],
[456, 396, 431, 494], [431, 397, 499, 496], [399, 477, 481, 412],
[429, 483, 478, 401], [410, 480, 490, 462], [496, 497, 484, 431],
[489, 495, 491, 459], [495, 460, 459, 471], [460, 489, 498, 498],
[472, 472, 471, 491]]
C_r.table == table3
C_c.table == table3
# Group denoted by B₂,₄ from [2] Pg. 474
F, a, b = free_group("a, b")
B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \
(a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4])
C_r = coset_enumeration_r(B_2_4, [a])
C_c = coset_enumeration_c(B_2_4, [a])
index_r = 0
for i in range(len(C_r.p)):
if C_r.p[i] == i:
index_r += 1
assert index_r == 1024
index_c = 0
for i in range(len(C_c.p)):
if C_c.p[i] == i:
index_c += 1
assert index_c == 1024
# trivial Macdonald group G(2,2) from [2] Pg. 480
M = FpGroup(F, [
b**-1 * a**-1 * b * a * b**-1 * a * b * a**-2,
a**-1 * b**-1 * a * b * a**-1 * b * a * b**-2
])
C_r = coset_enumeration_r(M, [a])
C_r.compress()
C_r.standardize()
C_c = coset_enumeration_c(M, [a])
C_c.compress()
C_c.standardize()
table4 = [[0, 0, 0, 0]]
assert C_r.table == table4
assert C_c.table == table4
def test_scan_1():
# Example 5.1 from [1]
F, x, y = free_group("x, y")
f = FpGroup(F, [x**3, y**3, x**-1 * y**-1 * x * y])
c = CosetTable(f, [x])
c.scan_and_fill(0, x)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, x**3)
assert c.table == [[0, 0, None, None]]
assert c.p == [0]
assert c.n == 1
assert c.omega == [0]
c.scan_and_fill(0, y**3)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [None, None, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, x**-1 * y**-1 * x * y)
assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [2, 2, 0, 1]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, y**3)
assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \
[4, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(1, x**-1 * y**-1 * x * y)
assert c.table == [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], \
[None, 1, None, None], [1, 3, None, None]]
assert c.p == [0, 1, 2, 1, 1]
assert c.n == 3
assert c.omega == [0, 1, 2]
# Example 5.2 from [1]
f = FpGroup(F, [x**2, y**3, (x * y)**3])
c = CosetTable(f, [x * y])
c.scan_and_fill(0, x * y)
assert c.table == [[1, None, None, 1], [None, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, x**2)
assert c.table == [[1, 1, None, 1], [0, 0, 0, None]]
assert c.p == [0, 1]
assert c.n == 2
assert c.omega == [0, 1]
c.scan_and_fill(0, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(0, (x * y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, y**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]]
assert c.p == [0, 1, 2]
assert c.n == 3
assert c.omega == [0, 1, 2]
c.scan_and_fill(1, (x * y)**3)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 4, 1, 0],
[None, 2, 4, None], [2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 4]
assert c.n == 5
assert c.omega == [0, 1, 2, 3, 4]
c.scan_and_fill(2, x**2)
assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 3, 1, 0], [2, 2, 3, 3],
[2, None, None, 3]]
assert c.p == [0, 1, 2, 3, 3]
assert c.n == 4
assert c.omega == [0, 1, 2, 3]
def test_permutation_methods():
from sympy.combinatorics.fp_groups import FpSubgroup
F, x, y = free_group("x, y")
# DihedralGroup(8)
G = FpGroup(F, [x**2, y**8, x * y * x**-1 * y])
T = G._to_perm_group()[1]
assert T.is_isomorphism()
assert G.center() == [y**4]
# DiheadralGroup(4)
G = FpGroup(F, [x**2, y**4, x * y * x**-1 * y])
S = FpSubgroup(G, G.normal_closure([x]))
assert x in S
assert y**-1 * x * y in S
# Z_5xZ_4
G = FpGroup(F, [x * y * x**-1 * y**-1, y**5, x**4])
assert G.is_abelian
assert G.is_solvable
# AlternatingGroup(5)
G = FpGroup(F, [x**3, y**2, (x * y)**5])
assert not G.is_solvable
# AlternatingGroup(4)
G = FpGroup(F, [x**3, y**2, (x * y)**3])
assert len(G.derived_series()) == 3
S = FpSubgroup(G, G.derived_subgroup())
assert S.order() == 4
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_simplify_presentation():
# ref #16083
G = simplify_presentation(FpGroup(FreeGroup([]), []))
assert not G.generators
assert not G.relators
