View source for Baer group

A Baer subplane in a [[projective plane]] is a subplane with the property that any point of the plane is incident with a line of the subplane and any line of the plane intersects the subplane in at least one point. For finite projective planes, the order of the plane is $n$ when there are exactly $n+1$ points per line. A subplane of order $k$ in a projective plane of order $n$ is Baer if and only if $k^2=n$. A Baer group is a collineation [[group]] of a projective plane which fixes each point of a Baer subplane. It is possible to use Baer groups to characterize those planes which correspond to hyperbolic and conical [[flock]]s. If the flock is in $\mathrm{PG}(3,K)$, for $K$ a [[field]], a Baer subplane fixed pointwise by a Baer group is a line of $\mathrm{PG}(3,K)$ which is not in the spread. A Baer group is maximal of elation, respectively homology, type if the group acts transitively on the non-fixed points on each line $L$ of the spread which intersects the Baer subplane and fixes one, respectively two, point(s) of $L$.

The translation planes with spread in $\mathrm{PG}(3,K)$ that admit maximal Baer groups of elation or homology type are in one-to-one correspondence with partial conical or hyperbolic flocks of deficiency one, respectively (see [[Flock]] (also for additional references); [[#References|[a1]]], [[#References|[a2]]]).

====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,   "Flocks of hyperbolic quadrics and translation planes admitting affine homologies"  ''J. Geom.'' , '''34'''  (1989)  pp. 50–73</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  V. Jha,   N.L. Johnson,   "Structure theory for point-Baer and line-Baer collineation groups in affine planes" , ''Proc. Amer. Math. Soc. Conf. Iowa City''  (l996)</TD></TR>
</table>

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