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# 7 Determining the Isomorphism Class of Projective Planes

### Sections

1. Isomorphisms and Collineations
2. Central Collineations
3. Collineations on Baer Subplanes
4. Invariants for Projective Planes

The methods in this chapter do not deal with relative difference sets. Instead, they help studying projective planes. So if you have a relative difference set, you must first generate the projective plane it defines (if it does).

Projective planes are always assumed to consist of positive integers (as points) and sets of integers (as blocks). The incidence relation is assumed to be the element relation. The blocks of a projective plane must be sets.

The following methods generate a record characterising the projective plane. As most of the functions in this chapter need this data, the record returned by `ElationPrecalc` or `ElationPrecalcSmall` is the recommended representation of projective planes.

- `ElationPrecalc(` blocks`) F`
- `ElationPrecalcSmall(` blocks`) F`

  Given the blocks blocks of a projective plane, `ElationPrecalc(` blocks`)` returns a record conatining

  .points
    the points of the projective plane (immutable)
  .blocks
    the blocks as passed to the function (immutable)
  .jpoint
    a matrix with ij-th entry the point meeting the i-th and the j-th block.
  .jblock
    a matrix with ij-th entry the position of the block connecting the point i to the point j in blocks.

  `ElationPrecalcSmall(` blocks`)` returns a record which does only contain .points, .blocks and .jblock. Hence the name.

  In the following sections, some of the functions have two versions. The versions which have a `Small` appended to it's name do not depend on the data generated by `ElationPrecalc`, but rather on the data structure provided by `ElationPrecalcSmall`. The `Small` versions are generally much slower than the other ones.

- `DualPlane(` blocks`) O`

  For a projective plane given by blocks, `DualPlane(` blocks`)` returns a record containing a set of blocks defining the dual plane and a List image containing the same blocks such that image[p] is the image of the point p under duality. It is not tested, if the design defined by blocks is actually a projective plane.

- `ProjectiveClosureOfPointSet(` points`,` maxsize`,` data`) O`

  Let P be a projective plane given by the record data as returned by `ElationPrecalcSmall`. Let points be a set of points (integers). Then `ProjectiveClosureOfPointSet` returns the projective colsure of points in P (the smallest subplane of P containing the points points). The closure is returned as a list of points. If maxsize ≠ 0, calculations are stopped if the closure is known to have at least maxsize points and data.points is returned. Observe that this is a ``small'' function, in the sense that it does not need the data from `ElationPrecalc` but merely the data generated by `ElationPrecalcSmall`.

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