Module orthogonal

Orthogonal Linear Groups

Paraphrased from the GAP manual: The general orthogonal group
$GO(e,d,q)$ consists of those $d\times d$ matrices over the field
$GF(q)$ that respect a non-singular quadratic form specified by
$e$. (Use the GAP command InvariantQuadraticForm to determine this
form explicitly.) The value of $e$ must be $0$ for odd $d$ (and can
optionally be omitted in this case), respectively one of $1$ or $-1$
for even $d$.

SpecialOrthogonalGroup returns a group isomorphic to the special 
orthogonal group $SO(e,d,q)$, which is the subgroup of all those 
matrices in the general orthogonal group that have determinant one. 
(The index of $SO(e,d,q)$ in $GO(e,d,q)$ is $2$ if $q$ is odd, 
but $SO(e,d,q) = GO(e,d,q)$ if $q$ is even.)
 
WARNING:
    GAP notation:  GO([e,] d, q), SO([e,] d, q)  ([...] denotes and optional value)
    
    SAGE notation: GO(d, GF(q), e=0), SO( d, GF(q), e=0)
    
    There is no Python trick I know of to allow the first argument to
    have the default value e=0 and leave the other two arguments as
    non-default.  This forces us into non-standard notation.
    
AUTHOR:
    -- David Joyner: initial version (2006-3)
    -- David Joyner (2006-05): added examples, _latex_, __str__,
                                     gens, as_matrix_group
    -- William Stein (2006-12-09): rewrite                                      

Return the special orthogonal group of degree $n$ over the ring $R$.

INPUT:
    n -- the degree
    R -- ring
    e -- a parameter for orthogonal groups only depending
             on the invariant form

EXAMPLES:
    sage: G = SO(3,GF(5))
    sage: G.gens()
    [
    [2 0 0]
    [0 3 0]
    [0 0 1],
    [3 2 3]
    [0 2 0]
    [0 3 1],
    [1 4 4]
    [4 0 0]
    [2 0 4]
    ]       
    sage: G = SO(3,GF(5))
    sage: G.as_matrix_group()
    Matrix group over Finite Field of size 5 with 3 generators:
    [[[2, 0, 0], [0, 3, 0], [0, 0, 1]], [[3, 2, 3], [0, 2, 0], [0, 3, 1]], [[1, 4, 4], [4, 0, 0], [2, 0, 4]]]

Return the general orthogonal group.

EXAMPLES: