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Difference between revisions of "Baer group"

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A Baer subplane in a [[Projective plane|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100301.png" /> when there are exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100302.png" /> points per line. A subplane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100303.png" /> in a projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100304.png" /> is Baer if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100305.png" />. A Baer group is a collineation [[Group|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 (cf. also [[Flock|Flock]]). If the flock is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100306.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100307.png" /> a [[Field|field]], a Baer subplane fixed pointwise by a Baer group is a line of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100308.png" /> which is not in the spread (cf. [[Flock|Flock]]). A Baer group is maximal of elation, respectively homology, type if the group acts transitively on the non-fixed points on each line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b1100309.png" /> of the spread which intersects the Baer subplane and fixes one, respectively two, point(s) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b11003010.png" />.
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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 [[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110030/b11003011.png" /> 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|Flock]] (also for additional references); [[#References|[a1]]], [[#References|[a2]]]).
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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====
 
====References====
<table><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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<table>
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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>
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<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>
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</table>
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Latest revision as of 17:24, 2 September 2017

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://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