コード例 #1
0
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
コード例 #2
0
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
コード例 #3
0
ファイル: test_homomorphisms.py プロジェクト: Lenqth/sympy
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
コード例 #4
0
ファイル: fp_groups.py プロジェクト: sixpearls/sympy
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
コード例 #5
0
ファイル: test_fp_groups.py プロジェクト: asmeurer/sympy
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)
コード例 #6
0
ファイル: test_fp_groups.py プロジェクト: asmeurer/sympy
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
コード例 #7
0
ファイル: coset_table.py プロジェクト: normalhuman/sympy
 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 = {}
コード例 #8
0
ファイル: test_fp_groups.py プロジェクト: alexako/sympy
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
コード例 #9
0
ファイル: test_fp_groups.py プロジェクト: bjodah/sympy
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
コード例 #10
0
ファイル: fp_groups.py プロジェクト: bjodah/sympy
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
コード例 #11
0
ファイル: test_fp_groups.py プロジェクト: asmeurer/sympy
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
コード例 #12
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ファイル: test_fp_groups.py プロジェクト: sixpearls/sympy
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
コード例 #13
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ファイル: test_homomorphisms.py プロジェクト: Lenqth/sympy
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)
コード例 #14
0
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
コード例 #15
0
ファイル: fp_groups.py プロジェクト: bjodah/sympy
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
コード例 #16
0
ファイル: fp_groups.py プロジェクト: KonstantinTogoi/sympy
    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
コード例 #17
0
ファイル: test_rewriting.py プロジェクト: sixpearls/sympy
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
コード例 #18
0
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
コード例 #19
0
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
コード例 #20
0
ファイル: test_rewriting.py プロジェクト: Lenqth/sympy
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
コード例 #21
0
ファイル: fp_groups.py プロジェクト: bjodah/sympy
    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
コード例 #22
0
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
コード例 #23
0
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]
コード例 #24
0
ファイル: test_fp_groups.py プロジェクト: vishalbelsare/sympy
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]
コード例 #25
0
ファイル: fp_groups.py プロジェクト: vishalbelsare/sympy
 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)
コード例 #26
0
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]
コード例 #27
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
コード例 #28
0
ファイル: test_fp_groups.py プロジェクト: hridog00/Proyecto
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]
コード例 #29
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
コード例 #30
0
ファイル: test_coset_table.py プロジェクト: asmeurer/sympy
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
コード例 #31
0
ファイル: fp_groups.py プロジェクト: sixpearls/sympy
 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)
コード例 #32
0
ファイル: test_coset_table.py プロジェクト: msgoff/sympy
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
コード例 #33
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
コード例 #34
0
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]
コード例 #35
0
ファイル: fp_groups.py プロジェクト: sixpearls/sympy
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
コード例 #36
0
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]
コード例 #37
0
ファイル: test_fp_groups.py プロジェクト: hridog00/Proyecto
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