Python SymmetricGroup.gens方法代碼示例

# 需要導入模塊: from sage.groups.perm_gps.permgroup_named import SymmetricGroup [as 別名]
# 或者: from sage.groups.perm_gps.permgroup_named.SymmetricGroup import gens [as 別名]
def SymmetricPresentation(n):
    r"""
    Build the Symmetric group of order `n!` as a finitely presented group.

    INPUT:

    - ``n`` -- The size of the underlying set of arbitrary symbols being acted
      on by the Symmetric group of order `n!`.

    OUTPUT:

    Symmetric group as a finite presentation, implementation uses GAP to find an
    isomorphism from a permutation representation to a finitely presented group
    representation. Due to this fact, the exact output presentation may not be
    the same for every method call on a constant ``n``.

    EXAMPLES::

        sage: S4 = groups.presentation.Symmetric(4)
        sage: S4.as_permutation_group().is_isomorphic(SymmetricGroup(4))
        True

    TESTS::

        sage: S = [groups.presentation.Symmetric(i) for i in range(1,4)]; S[0].order()
        1
        sage: S[1].order(), S[2].as_permutation_group().is_isomorphic(DihedralGroup(3))
        (2, True)
        sage: S5 = groups.presentation.Symmetric(5)
        sage: perm_S5 = S5.as_permutation_group(); perm_S5.is_isomorphic(SymmetricGroup(5))
        True
        sage: groups.presentation.Symmetric(8).order()
        40320
    """
    from sage.groups.perm_gps.permgroup_named import SymmetricGroup
    from sage.groups.free_group import _lexi_gen

    n = Integer(n)
    perm_rep = SymmetricGroup(n)
    GAP_fp_rep = libgap.Image(libgap.IsomorphismFpGroupByGenerators(perm_rep, perm_rep.gens()))
    image_gens = GAP_fp_rep.FreeGeneratorsOfFpGroup()
    name_itr = _lexi_gen()  # Python generator object for variable names
    F = FreeGroup([next(name_itr) for x in perm_rep.gens()])
    ret_rls = tuple(
        [F(rel_word.TietzeWordAbstractWord(image_gens).sage()) for rel_word in GAP_fp_rep.RelatorsOfFpGroup()]
    )
    return FinitelyPresentedGroup(F, ret_rls)

# 需要導入模塊: from sage.groups.perm_gps.permgroup_named import SymmetricGroup [as 別名]
# 或者: from sage.groups.perm_gps.permgroup_named.SymmetricGroup import gens [as 別名]
def CyclicCodeFromGeneratingPolynomial(n,g,ignore=True):
    r"""
    If g is a polynomial over GF(q) which divides `x^n-1` then
    this constructs the code "generated by g" (ie, the code associated
    with the principle ideal `gR` in the ring
    `R = GF(q)[x]/(x^n-1)` in the usual way).

    The option "ignore" says to ignore the condition that (a) the
    characteristic of the base field does not divide the length (the
    usual assumption in the theory of cyclic codes), and (b) `g`
    must divide `x^n-1`. If ignore=True, instead of returning
    an error, a code generated by `gcd(x^n-1,g)` is created.

    EXAMPLES::

        sage: P.<x> = PolynomialRing(GF(3),"x")
        sage: g = x-1
        sage: C = codes.CyclicCodeFromGeneratingPolynomial(4,g); C
        Linear code of length 4, dimension 3 over Finite Field of size 3
        sage: P.<x> = PolynomialRing(GF(4,"a"),"x")
        sage: g = x^3+1
        sage: C = codes.CyclicCodeFromGeneratingPolynomial(9,g); C
        Linear code of length 9, dimension 6 over Finite Field in a of size 2^2
        sage: P.<x> = PolynomialRing(GF(2),"x")
        sage: g = x^3+x+1
        sage: C = codes.CyclicCodeFromGeneratingPolynomial(7,g); C
        Linear code of length 7, dimension 4 over Finite Field of size 2
        sage: C.generator_matrix()
        [1 1 0 1 0 0 0]
        [0 1 1 0 1 0 0]
        [0 0 1 1 0 1 0]
        [0 0 0 1 1 0 1]
        sage: g = x+1
        sage: C = codes.CyclicCodeFromGeneratingPolynomial(4,g); C
        Linear code of length 4, dimension 3 over Finite Field of size 2
        sage: C.generator_matrix()
        [1 1 0 0]
        [0 1 1 0]
        [0 0 1 1]

    On the other hand, CyclicCodeFromPolynomial(4,x) will produce a
    ValueError including a traceback error message: "`x` must
    divide `x^4 - 1`". You will also get a ValueError if you
    type

    ::

        sage: P.<x> = PolynomialRing(GF(4,"a"),"x")
        sage: g = x^2+1

    followed by CyclicCodeFromGeneratingPolynomial(6,g). You will also
    get a ValueError if you type

    ::

        sage: P.<x> = PolynomialRing(GF(3),"x")
        sage: g = x^2-1
        sage: C = codes.CyclicCodeFromGeneratingPolynomial(5,g); C
        Linear code of length 5, dimension 4 over Finite Field of size 3

    followed by C = CyclicCodeFromGeneratingPolynomial(5,g,False), with
    a traceback message including "`x^2 + 2` must divide
    `x^5 - 1`".
    """
    P = g.parent()
    x = P.gen()
    F = g.base_ring()
    p = F.characteristic()
    if not(ignore) and p.divides(n):
        raise ValueError('The characteristic %s must not divide %s'%(p,n))
    if not(ignore) and not(g.divides(x**n-1)):
        raise ValueError('%s must divide x^%s - 1'%(g,n))
    gn = GCD([g,x**n-1])
    d = gn.degree()
    coeffs = Sequence(gn.list())
    r1 = Sequence(coeffs+[0]*(n - d - 1))
    Sn = SymmetricGroup(n)
    s = Sn.gens()[0] # assumes 1st gen of S_n is (1,2,...,n)
    rows = [permutation_action(s**(-i),r1) for i in range(n-d)]
    MS = MatrixSpace(F,n-d,n)
    return LinearCode(MS(rows))