A373501 Size of the collineation group of classical projective planes of prime power order q.

168, 5616, 120960, 372000, 5630688, 49448448, 84913920, 212427600, 810534816, 17108582400, 6950204928, 16934047920, 78156525216, 304668000000, 846083360304, 499631102880, 851974934400, 5492021821440, 3509844434208, 7980059337600, 11681731985616, 23800278205248

Offset: 1

Author

Ralf Goertz, Jun 07 2024

Keywords

• nonn

Comments

a(A246655(n)) is the size of the collineation group of the classical projective plane of order q=p^k. It is also known as the projective semilinear group, PGammaL(3,q), the semidirect product of PGL(3,q) (whose order is probably given by A003800) with the group of field automorphisms of F(q). The latter is the cyclic group of order k. Therefore, |PGammaL(3,p^k)|=|PGL(3,p^k)|*k.

Examples

			Take for example the first value 168 which refers to the number of automorphisms of the Fano plane (q=2). Its v=7 (=q^2+q+1) lines are subsets of size 3 (=q+1) of a set of v points. Using 0,1,...,6 to label these points, one way of enumerating the lines is depicted in the first column of the following table:
            (0 1 2 3 4 5 6)   (0 6)(3 5)
  {0,1,3}       {1,2,4}        {6,1,5}
  {1,2,4}       {2,3,5}        {1,2,4}
  {2,3,5}       {3,4,6}        {2,5,3}
  {3,4,6}       {4,5,0}        {5,4,0}
  {4,5,0}       {5,6,1}        {4,3,6}
  {5,6,1}       {6,0,2}        {3,0,1}
  {6,0,2}       {0,1,3}        {0,6,2}
Note that any two distinct lines have exactly 1 point in common. Applying one of the 7!=5040 possible permutations of the points obviously doesn't change that fact. However, exactly 168 of these permutations lead to the same set of subsets. One such permutation is the full cycle (0,1,2,3,4,5,6) whose action can bee seen in the second column. It also permutes the lines cyclically by mapping line i to line i+1 (mod v). Another one is the cycle product (0 6)(3 5) in the third column. It swaps lines 1 and 6 and lines 4 and 5 and leaves the other three lines fixed.
		

References

• A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.
• D. R. Hughes and F. C. Piper, Projective Planes, Springer, 1973.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Wikipedia, Projective linear group, order of PGL(3,q) and PGammaL(3,q).

Crossrefs

Cf. A373502 for the size of a complete set of classical projective planes using a given set of q^2+q+1 points.

Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.

Programs

Mathematica

Map[PrimeOmega[#]*#^3*(#^2+#+1)*(#^2-1)*(#-1) &, Select[Range[50], PrimePowerQ]] (* Paolo Xausa, Aug 01 2024 *)

PARI

a=(q)->bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) \\ q=A246655(n)

Formula

a(n) = Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) where q = A246655(n).

Extensions

Data corrected by Paolo Xausa, Aug 02 2024