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Baer group

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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 flocks. 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); [a1], [a2]).

References

[a1] N.L. Johnson, "Flocks of hyperbolic quadrics and translation planes admitting affine homologies" J. Geom. , 34 (1989) pp. 50–73
[a2] 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)
How to Cite This Entry:
Baer group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Baer_group&oldid=41784
This article was adapted from an original article by N.J. Johnson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article