Orthogonal Linear Groups with GAP - Groups

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# Orthogonal Linear Groups with GAP

class sage.groups.matrix_gps.orthogonal_gap.OrthogonalMatrixGroup_gap(degree, base_ring, special, sage_name, latex_string, gap_command_string, category=None)

  Bases: `OrthogonalMatrixGroup_generic`, `NamedMatrixGroup_gap`, `FinitelyGeneratedMatrixGroup_gap`

  The general or special orthogonal group in GAP.

  invariant_bilinear_form()

    Return the symmetric bilinear form preserved by the orthogonal group.

    OUTPUT:

    A matrix \(M\) such that, for every group element \(g\), the identity \(g m g^T = m\) holds. In characteristic different from two, this uniquely determines the orthogonal group.

    EXAMPLES:

    Sage

        sage: G = GO(4, GF(7), -1)
        sage: G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 2 0]
        [0 0 0 2]

        sage: G = GO(4, GF(7), +1)
        sage: G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 6 0]
        [0 0 0 2]

        sage: G = SO(4, GF(7), -1)
        sage: G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 2 0]
        [0 0 0 2]

    Python

        >>> from sage.all import *
        >>> G = GO(Integer(4), GF(Integer(7)), -Integer(1))
        >>> G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 2 0]
        [0 0 0 2]

        >>> G = GO(Integer(4), GF(Integer(7)), +Integer(1))
        >>> G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 6 0]
        [0 0 0 2]

        >>> G = SO(Integer(4), GF(Integer(7)), -Integer(1))
        >>> G.invariant_bilinear_form()
        [0 1 0 0]
        [1 0 0 0]
        [0 0 2 0]
        [0 0 0 2]

  invariant_form()

    alias of `invariant_bilinear_form()`.

  invariant_quadratic_form()

    Return the quadratic form preserved by the orthogonal group.

    OUTPUT:

    The matrix \(Q\) defining “orthogonal” as follows. The matrix determines a quadratic form \(q\) on the natural vector space \(V\), on which \(G\) acts, by \(q(v) = v Q v^t\). A matrix \(M\) is an element of the orthogonal group if \(q(v) = q(v M)\) for all \(v \in V\).

    EXAMPLES:

    Sage

        sage: G = GO(4, GF(7), -1)
        sage: G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 1 0]
        [0 0 0 1]

        sage: G = GO(4, GF(7), +1)
        sage: G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 3 0]
        [0 0 0 1]

        sage: G = GO(4, QQ)
        sage: G.invariant_quadratic_form()
        [1 0 0 0]
        [0 1 0 0]
        [0 0 1 0]
        [0 0 0 1]

        sage: G = SO(4, GF(7), -1)
        sage: G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 1 0]
        [0 0 0 1]

    Python

        >>> from sage.all import *
        >>> G = GO(Integer(4), GF(Integer(7)), -Integer(1))
        >>> G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 1 0]
        [0 0 0 1]

        >>> G = GO(Integer(4), GF(Integer(7)), +Integer(1))
        >>> G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 3 0]
        [0 0 0 1]

        >>> G = GO(Integer(4), QQ)
        >>> G.invariant_quadratic_form()
        [1 0 0 0]
        [0 1 0 0]
        [0 0 1 0]
        [0 0 0 1]

        >>> G = SO(Integer(4), GF(Integer(7)), -Integer(1))
        >>> G.invariant_quadratic_form()
        [0 1 0 0]
        [0 0 0 0]
        [0 0 1 0]
        [0 0 0 1]