There are two classes of such subgroups: the stabilizers of a point
(and a plane) of and the stabilizers of an isotropic line of
.
There are two such classes: the stabilizers
of a pair of isotropic lines (if ) and the stabilizers of a pair of
non-isotropic lines.
There are two such classes:
A subgroup in 3(a)i fixes as a whole the set of lines of
intersecting both
and
, when seen as lines in
. Note that
and
correspond to each other in the symplectic polarity associated to the symplectic
form
.
Hence all the lines intersecting
and
are isotropic (see
[Bert2]) forming in
an
elliptic linear congruence contained in the complex
of isotropic lines.
In the orthogonal representation such a congruence is a quadric of index 1
lying on
, and therefore these subgroups are isomorphic to
.
In the symplectic representation they appear with the obvious structure
(see [KL]).
A subgroup in 3(a)ii leaves invariant the set of all lines of
through
and
. These lines form in
an elliptic linear congruence
which is not contained in
.
The isotropic lines of
form
a regulus, and therefore determine
a quadric of
which is fixed by the relevant subgroup in
3(a)ii.
In the orthogonal representation, the quadric is a conic on
, and
these subgroups are therefore
isomorphic to
.
In the symplectic representation they
are easily described as having structure
(see [KL]).
There is a unique class of subgroups of type 3(b)i under , whereas
groups of type 3(b)ii are all conjugate under
.
These subgroups appear in the list when is a prime. Their preimages in
have structure
. In
their structure is
if
(in which
case we have two classes)
and
if
(in which case we have one class)
(see [Asch2] or [KL]).
In the orthogonal representation these subgroups have a natural description:
they are the subgroups of
stabilizing a simplex of
(thus
monomial on
).
Given a twisted cubic
in
it is easily seen (cf. [SK]) that
the correspondence that sends a point of
to the osculating plane
through the point extends to a symplectic polarity.
Conversely given a
symplectic polarity in
, say
, a twisted cubic can be
associated to
in such a way that the correspondence point-osculating
plane through the point extends to
.
E.g., if
is associated to the symplectic
form
of matrix
the cubic having parametric equation
,
, meets the condition above. Indeed,
explicit computation shows that if
is a cubic associated
to a polarity of matrix
, the cubic
,
image
of
via a transformation of matrix
, is associated to a polarity of matrix
,
where
for some constant
in the field. Hence for a given
, two cubics are associated to
if
and only if they can be transformed into one another by an element of
.
If ,
the stabilizer of such a cubic in
is
a maximal subgroup (see [Mit]); this subgroup is isomorphic to
and its action on the points of the
cubic is permutationally equivalent to the natural action on the
projective line (see [SK]
and [L\"un]).
There are three classes of maximal subgroups isomorphic respectively to
(if
is a prime
);
(if
is a prime
,
) and
(if
).