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[CPP: 55.38] can be improved: Two closed Baer subplanes always have a point and a line in common, not only if they are the fixed point sets of involutions. See R. Löwen [1998b].

[CPP: 65.2] can be improved: it suffices to assume that
dimS ³ C_{l}-2.
The following additional argument is then required in step 1) of the
*proof*:

If B is contained in a line L through x,
then S_{x}=S_{x,L} and, for a
suitable s Î S, the stabilizer S_{x,xs} fixes a
triangle, but dimS_{x,xs} ³ C_{l}-2-4*l* is too large.

**8.a.**

The following results improve [CPP: 87.1]:

*If P is a 16-dimensional plane, and if dimD ³ 27,
then D is a Lie group. *

See B. Priwitzer - H. Salzmann [1998].

*Irrespective of connectedness, the automorphism groups of a
16-dimensional projective plane is a Lie group, if it has dimension at
least 29*; see
H. Salzmann [1999a].

**8.b.**

Regarding [CPP: 87.2] we remark that the same results have been obtained
in
B. Priwitzer [1997]
and
B. Priwitzer [1998]
under the weaker hypothesis that dimD > 28 (instead of 30).

**8.c.**

[CPP: 87.3] is true for dimD ³ 31 instead of 35.
See
H. Salzmann [1998].

**8.d.**

ad [CPP: 87.4]: the arguments on p.587 can be improved as follows:

**Theorem.** *If D is not semi-simple, and if dimD ³ 33, then, up to duality,
D has a minimal normal subgroup Q @ R^{t} consisting of
axial collineations with common axis W. *

Either Q £ D_{[a,W]} is a group of homologies and t=1, or Q is contained in the group T = D_{[W,W]} of elations with axis W.

For full proofs see H. Salzmann [1999b].

**8.e.**

The results described in Section 87 have been extended, as follows.

**Theorem.** *If dimD ³ 35 and if D fixes exactly one
line W and no point, then P is a translation plane. *

Recently, Hähl and Löwe have determined explicitly all translation planes having a group D as in this theorem. In particular, their work implies the following:

If D fixes two distinct points, and if dimD = 39, then

**8.f.**

The following results are obtained in
H. Salzmann [1999d]:

**Lemma.** *Assume that dimD ³ 29. If D fixes no
point and no line, then P is
classical, or D¢ @ SL_{3}(H) and P is a
Hughes plane. If D fixes exactly one element,
then D has a normal vector subgroup. *

**8.g.**

Moreover, the following is true
(H. Salzmann [2000-2003]):

**Theorem.** *If dimD ³ 34 and if
D fixes exactly 2 points u,v and the line
uv, then the group T of translations with axis uv
is at least 15-dimensional. Either D has a subgroup
Y @ Spin _{7} R and
dimD ³ 36, or T is transitive, a maximal semi-simple
subgroup of
D is isomorphic to SU_{4} C @ Spin_{6}R,
and dimD = 34 *.

**Theorem.** *If D fixes exactly 2 points u,v and
2 lines W=uv and Y=av,
then the translation group T=D _{[v,W]} is transitive, the
complement D_{a}
of
T has a compact commutator group F @ Spin_{8} R, and
dimD ³ 36. If even dimD ³ 38, then P is the classical Moufang plane *.

**Theorem.** *If dimD ³ 32 and D has * (*
at least *) *3 fixed points, then
D contains a transitive translation group T. Either
dimD = 32 and a maximal semi-simple
subgroup Y of D is isomorphic to
SU _{4} C, or dimD ³ 37 and P is the
classical Moufang plane *.

File translated from T

On 16 Jan 2003, 14:57.