[Up] [Previous] [Next] [Index]

# 7.3 Collineations on Baer Subplanes

Let P be a projective plane of order n2. A subplane B of order n of P is called Baer subplane. Baer suplanes are exactly the maximal subplanes of P.

- `InducedCollineation(` baerdata`,` baercoll`,` point`,` image`,` planedata`,` liftingperm`) O`

  If a projective plane contains a Baer subplane, collineations of the subplane may be lifted to the full plane. Here baercoll is a collineation of the subplane given by baerdata (as returned by `ElationPrecalc`. Be careful, as the enumeration for the subplane is not the same as for the whole plane). liftingperm is a permutation on the points of the full pane which converts the enumeration of the subplane to that of the full plane. This means that the image of baerdata.points under liftingperm is a subset of planedata.points. Namely the one representing the Baer plane in the enumeration used for the whole plane. point and image are points outside the Baer plane.

  `InducedCollineation` returns a collineation of the full plane (as a permutation on planedata.points) which takes point to image and acts on the Baer plane as baercoll does.

  Just to make this clear again, baerdata has points [1,...,n2+n+1] and planedata has points [1,...,n4+n2+1]. baercoll lives on baerdata.points (and hence on n2+n+1 points) and point and image live on planedata.points. Anything can happen if you mix something up here.

  [Up] [Previous] [Next] [Index]