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_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():
# 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 define_schreier_generators(C):
y = []
gamma = 1
f = C.fp_group
X = f.generators
C.P = [[None]*len(C.A) for i in range(C.n)]
for alpha, x in product(C.omega, C.A):
beta = C.table[alpha][C.A_dict[x]]
if beta == gamma:
C.P[alpha][C.A_dict[x]] = "<identity>"
C.P[beta][C.A_dict_inv[x]] = "<identity>"
gamma += 1
elif x in X and C.P[alpha][C.A_dict[x]] is None:
y_alpha_x = '%s_%s' % (x, alpha)
y.append(y_alpha_x)
C.P[alpha][C.A_dict[x]] = y_alpha_x
grp_gens = list(free_group(', '.join(y)))
C._schreier_free_group = grp_gens.pop(0)
C._schreier_generators = grp_gens
# replace all elements of P by, free group elements
for i, j in product(range(len(C.P)), range(len(C.A))):
# if equals "<identity>", replace by identity element
if C.P[i][j] == "<identity>":
C.P[i][j] = C._schreier_free_group.identity
elif isinstance(C.P[i][j], str):
r = C._schreier_generators[y.index(C.P[i][j])]
C.P[i][j] = r
beta = C.table[i][j]
C.P[beta][j + 1] = r**-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_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 __init__(self, fp_grp, subgroup, max_cosets=None):
if not max_cosets:
max_cosets = CosetTable.coset_table_max_limit
self.fp_group = fp_grp
self.subgroup = subgroup
self.coset_table_max_limit = max_cosets
# "p" is setup independent of Ω and n
self.p = [0]
# a list of the form `[gen_1, gen_1^{-1}, ... , gen_k, gen_k^{-1}]`
self.A = list(chain.from_iterable((gen, gen**-1) \
for gen in self.fp_group.generators))
#P[alpha, x] Only defined when alpha^x is defined.
self.P = [[None]*len(self.A)]
# the mathematical coset table which is a list of lists
self.table = [[None]*len(self.A)]
self.A_dict = {x: self.A.index(x) for x in self.A}
self.A_dict_inv = {}
for x, index in self.A_dict.items():
if index % 2 == 0:
self.A_dict_inv[x] = self.A_dict[x] + 1
else:
self.A_dict_inv[x] = self.A_dict[x] - 1
# used in the coset-table based method of coset enumeration. Each of
# the element is called a "deduction" which is the form (α, x) whenever
# a value is assigned to α^x during a definition or "deduction process"
self.deduction_stack = []
# Attributes for modified methods.
H = self.subgroup
self._grp = free_group(', ' .join(["a_%d" % i for i in range(len(H))]))[0]
self.P = [[None]*len(self.A)]
self.p_p = {}
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_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 simplify_presentation(*args, **kwargs):
'''
For an instance of `FpGroup`, return a simplified isomorphic copy of
the group (e.g. remove redundant generators or relators). Alternatively,
a list of generators and relators can be passed in which case the
simplified lists will be returned.
By default, the generators of the group are unchanged. If you would
like to remove redundant generators, set the keyword argument
`change_gens = True`.
'''
change_gens = kwargs.get("change_gens", False)
if len(args) == 1:
if not isinstance(args[0], FpGroup):
raise TypeError("The argument must be an instance of FpGroup")
G = args[0]
gens, rels = simplify_presentation(G.generators, G.relators,
change_gens=change_gens)
if gens:
return FpGroup(gens[0].group, rels)
return FpGroup(FreeGroup([]), [])
elif len(args) == 2:
gens, rels = args[0][:], args[1][:]
if not gens:
return gens, rels
identity = gens[0].group.identity
else:
if len(args) == 0:
m = "Not enough arguments"
else:
m = "Too many arguments"
raise RuntimeError(m)
prev_gens = []
prev_rels = []
while not set(prev_rels) == set(rels):
prev_rels = rels
while change_gens and not set(prev_gens) == set(gens):
prev_gens = gens
gens, rels = elimination_technique_1(gens, rels, identity)
rels = _simplify_relators(rels, identity)
if change_gens:
syms = [g.array_form[0][0] for g in gens]
F = free_group(syms)[0]
identity = F.identity
gens = F.generators
subs = dict(zip(syms, gens))
for j, r in enumerate(rels):
a = r.array_form
rel = identity
for sym, p in a:
rel = rel*subs[sym]**p
rels[j] = rel
return gens, rels
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_free_group():
G, a, b, c = free_group("a, b, c")
assert F.generators == (x, y, z)
assert x*z**2 in F
assert x in F
assert y*z**-1 in F
assert (y*z)**0 in F
assert a not in F
assert a**0 not in F
assert len(F) == 3
assert str(F) == '<free group on the generators (x, y, z)>'
assert not F == G
assert F.order() == oo
assert F.is_abelian == False
assert F.center() == set([F.identity])
(e,) = free_group("")
assert e.order() == 1
assert e.generators == ()
assert e.elements == set([e.identity])
assert e.is_abelian == True
def define_schreier_generators(C, homomorphism=False):
'''
Parameters
==========
C -- Coset table.
homomorphism -- When set to True, return a dictionary containing the images
of the presentation generators in the original group.
'''
y = []
gamma = 1
f = C.fp_group
X = f.generators
if homomorphism:
# `_gens` stores the elements of the parent group to
# to which the schreier generators correspond to.
_gens = {}
# compute the schreier Traversal
tau = {}
tau[0] = f.identity
C.P = [[None]*len(C.A) for i in range(C.n)]
for alpha, x in product(C.omega, C.A):
beta = C.table[alpha][C.A_dict[x]]
if beta == gamma:
C.P[alpha][C.A_dict[x]] = "<identity>"
C.P[beta][C.A_dict_inv[x]] = "<identity>"
gamma += 1
if homomorphism:
tau[beta] = tau[alpha]*x
elif x in X and C.P[alpha][C.A_dict[x]] is None:
y_alpha_x = '%s_%s' % (x, alpha)
y.append(y_alpha_x)
C.P[alpha][C.A_dict[x]] = y_alpha_x
if homomorphism:
_gens[y_alpha_x] = tau[alpha]*x*tau[beta]**-1
grp_gens = list(free_group(', '.join(y)))
C._schreier_free_group = grp_gens.pop(0)
C._schreier_generators = grp_gens
if homomorphism:
C._schreier_gen_elem = _gens
# replace all elements of P by, free group elements
for i, j in product(range(len(C.P)), range(len(C.A))):
# if equals "<identity>", replace by identity element
if C.P[i][j] == "<identity>":
C.P[i][j] = C._schreier_free_group.identity
elif isinstance(C.P[i][j], string_types):
r = C._schreier_generators[y.index(C.P[i][j])]
C.P[i][j] = r
beta = C.table[i][j]
C.P[beta][j + 1] = r**-1
def subgroup(self, gens, C=None):
'''
Return the subgroup generated by `gens` using the
Reidemeister-Schreier algorithm
'''
if not all([isinstance(g, FreeGroupElement) for g in gens]):
raise ValueError("Generators must be `FreeGroupElement`s")
if not all([g.group == self.free_group for g in gens]):
raise ValueError("Given generators are not members of the group")
g, rels = reidemeister_presentation(self, gens, C=C)
if g:
g = FpGroup(g[0].group, rels)
else:
g = FpGroup(free_group('')[0], [])
return g
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_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_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 subgroup(self, gens, C=None, homomorphism=False):
'''
Return the subgroup generated by `gens` using the
Reidemeister-Schreier algorithm
homomorphism -- When set to True, return a dictionary containing the images
of the presentation generators in the original group.
Examples
========
>>> from sympy.combinatorics.fp_groups import (FpGroup, FpSubgroup)
>>> from sympy.combinatorics.free_groups import free_group
>>> 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]
>>> K, T = f.subgroup(H, homomorphism=True)
>>> T(K.generators)
[x*y, x**-1*y**2*x**-1]
'''
if not all([isinstance(g, FreeGroupElement) for g in gens]):
raise ValueError("Generators must be `FreeGroupElement`s")
if not all([g.group == self.free_group for g in gens]):
raise ValueError("Given generators are not members of the group")
if homomorphism:
g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True)
else:
g, rels = reidemeister_presentation(self, gens, C=C)
if g:
g = FpGroup(g[0].group, rels)
else:
g = FpGroup(free_group('')[0], [])
if homomorphism:
from sympy.combinatorics.homomorphisms import homomorphism
return g, homomorphism(g, self, g.generators, _gens, check=False)
return g
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_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 to_FpGroup(self):
if isinstance(self.parent, FreeGroup):
gen_syms = [('x_%d' % i) for i in range(len(self.generators))]
return free_group(', '.join(gen_syms))[0]
return self.parent.subgroup(C=self.C)
def test_FreeGroup__eq__():
assert free_group("x, y, z")[0] == free_group("x, y, z")[0]
assert free_group("x, y, z")[0] is free_group("x, y, z")[0]
assert free_group("x, y, z")[0] != free_group("a, x, y")[0]
assert free_group("x, y, z")[0] is not free_group("a, x, y")[0]
assert free_group("x, y")[0] != free_group("x, y, z")[0]
assert free_group("x, y")[0] is not free_group("x, y, z")[0]
assert free_group("x, y, z")[0] != free_group("x, y")[0]
assert free_group("x, y, z")[0] is not free_group("x, y")[0]
from sympy.combinatorics.free_groups import free_group, FreeGroup
from sympy.core import Symbol
from sympy.utilities.pytest import raises
from sympy import oo
F, x, y, z = free_group("x, y, z")
def test_FreeGroup__init__():
x, y, z = map(Symbol, "xyz")
assert len(FreeGroup("x, y, z").generators) == 3
assert len(FreeGroup(x).generators) == 1
assert len(FreeGroup(("x", "y", "z"))) == 3
assert len(FreeGroup((x, y, z)).generators) == 3
def test_free_group():
G, a, b, c = free_group("a, b, c")
assert F.generators == (x, y, z)
assert x * z**2 in F
assert x in F
assert y * z**-1 in F
assert (y * z)**0 in F
assert a not in F
assert a**0 not in F
assert len(F) == 3
assert str(F) == '<free group on the generators (x, y, z)>'
assert not F == G
assert F.order() == oo
assert F.is_abelian == False
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]
from sympy.combinatorics.free_groups import free_group, FreeGroup
from sympy.core import Symbol
from sympy.utilities.pytest import raises
from sympy import oo
F, x, y, z = free_group("x, y, z")
def test_FreeGroup__init__():
x, y, z = map(Symbol, "xyz")
assert len(FreeGroup("x, y, z").generators) == 3
assert len(FreeGroup(x).generators) == 1
assert len(FreeGroup(("x", "y", "z"))) == 3
assert len(FreeGroup((x, y, z)).generators) == 3
def test_free_group():
G, a, b, c = free_group("a, b, c")
assert F.generators == (x, y, z)
assert x*z**2 in F
assert x in F
assert y*z**-1 in F
assert (y*z)**0 in F
assert a not in F
assert a**0 not in F
assert len(F) == 3
assert str(F) == '<free group on the generators (x, y, z)>'
assert not F == G
assert F.order() == oo
assert F.is_abelian == False
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 to_FpGroup(self):
if isinstance(self.parent, FreeGroup):
gen_syms = [('x_%d'%i) for i in range(len(self.generators))]
return free_group(', '.join(gen_syms))[0]
return self.parent.subgroup(C=self.C)
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 reidemeister_presentation(fp_grp, H, C=None):
"""
fp_group: A finitely presented group, an instance of FpGroup
H: A subgroup whose presentation is to be found, given as a list
of words in generators of `fp_grp`
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation
>>> F, x, y = free_group("x, y")
Example 5.6 Pg. 177 from [1]
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
>>> H = [x*y, x**-1*y**-1*x*y*x]
>>> reidemeister_presentation(f, H)
((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))
Example 5.8 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3, y**3, (x*y)**3])
>>> H = [x*y, x*y**-1]
>>> reidemeister_presentation(f, H)
((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))
Exercises Q2. Pg 187 from [1]
>>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**4,))
Example 5.9 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**6,))
"""
if not C:
C = coset_enumeration_r(fp_grp, H)
C.compress()
C.standardize()
define_schreier_generators(C)
reidemeister_relators(C)
prev_gens = []
prev_rels = []
while not set(prev_rels) == set(C._reidemeister_relators):
prev_rels = C._reidemeister_relators
while not set(prev_gens) == set(C._schreier_generators):
prev_gens = C._schreier_generators
elimination_technique_1(C)
simplify_presentation(C)
syms = [g.array_form[0][0] for g in C._schreier_generators]
g = free_group(syms)[0]
subs = dict(zip(syms, g.generators))
C._schreier_generators = g.generators
for j, r in enumerate(C._reidemeister_relators):
a = r.array_form
rel = g.identity
for sym, p in a:
rel = rel * subs[sym]**p
C._reidemeister_relators[j] = rel
C.schreier_generators = tuple(C._schreier_generators)
C.reidemeister_relators = tuple(C._reidemeister_relators)
return C.schreier_generators, C.reidemeister_relators
def test_FreeGroup__eq__():
assert free_group("x, y, z")[0] == free_group("x, y, z")[0]
assert free_group("x, y, z")[0] is free_group("x, y, z")[0]
assert free_group("x, y, z")[0] != free_group("a, x, y")[0]
assert free_group("x, y, z")[0] is not free_group("a, x, y")[0]
assert free_group("x, y")[0] != free_group("x, y, z")[0]
assert free_group("x, y")[0] is not free_group("x, y, z")[0]
assert free_group("x, y, z")[0] != free_group("x, y")[0]
assert free_group("x, y, z")[0] is not free_group("x, y")[0]
def reidemeister_presentation(fp_grp, H, C=None):
"""
fp_group: A finitely presented group, an instance of FpGroup
H: A subgroup whose presentation is to be found, given as a list
of words in generators of `fp_grp`
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation
>>> F, x, y = free_group("x, y")
Example 5.6 Pg. 177 from [1]
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
>>> H = [x*y, x**-1*y**-1*x*y*x]
>>> reidemeister_presentation(f, H)
((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))
Example 5.8 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3, y**3, (x*y)**3])
>>> H = [x*y, x*y**-1]
>>> reidemeister_presentation(f, H)
((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))
Exercises Q2. Pg 187 from [1]
>>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**4,))
Example 5.9 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**6,))
"""
if not C:
C = coset_enumeration_r(fp_grp, H)
C.compress(); C.standardize()
define_schreier_generators(C)
reidemeister_relators(C)
prev_gens = []
prev_rels = []
while not set(prev_rels) == set(C._reidemeister_relators):
prev_rels = C._reidemeister_relators
while not set(prev_gens) == set(C._schreier_generators):
prev_gens = C._schreier_generators
elimination_technique_1(C)
simplify_presentation(C)
syms = [g.array_form[0][0] for g in C._schreier_generators]
g = free_group(syms)[0]
subs = dict(zip(syms,g.generators))
C._schreier_generators = g.generators
for j, r in enumerate(C._reidemeister_relators):
a = r.array_form
rel = g.identity
for sym, p in a:
rel = rel*subs[sym]**p
C._reidemeister_relators[j] = rel
C.schreier_generators = tuple(C._schreier_generators)
C.reidemeister_relators = tuple(C._reidemeister_relators)
return C.schreier_generators, C.reidemeister_relators
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_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
