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User: Ralf Goertz

Ralf Goertz has authored 2 sequences.

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

Ralf Goertz, Jun 07 2024

nonn

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.

			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.

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.

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

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.

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

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

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

Data corrected by Paolo Xausa, Aug 02 2024

A373502 Size of a complete set of classical projective planes of prime power order q using a given set of q^2+q+1 points.

Original entry on oeis.org

30, 1108800, 422378820864000, 22104404984349254886359040000, 7197507570101063450093594584788274920397007398780859842560000000000000, 90399509839271079668491458784005740889517921781547218950513473999637402251071324160000000000000000

Offset: 1

Ralf Goertz, Jun 08 2024

nonn

Projective planes of order q can be seen as a set of v=q^2+q+1 subsets (the lines) of size q+1 of a set of v points with the property that any two distinct lines have exactly one point in common. Obviously, this also holds for any of the v! permutations of the points. However, some of these permutations map the points of a given line l of the plane to the points of another line l' thereby fixing the set of lines and consequently the whole projective plane. These permutations form a subgroup called the collineation group of the projective plane. The size of this group for classical projective planes is given by A373501. Therefore, a(q) is the index of the collineation subgroup in the symmetric group of the points where q=A246655(n).

			For the Fano plane (q=2) there are 7 points and 7 lines. Of the 7!=5040 permutations of the points 168 fix the set of lines and thereby the whole plane. Consequently, there are 5040/168=30 different such planes for any given set of points. See A373501 for a more elaborate discussion of this example.

A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.

Paolo Xausa, Table of n, a(n) for n = 1..12
IBM Research, Possible Dobble decks, Ponder This Challenge May 2024, asked for a(3) and a(6).
Wikipedia, Order of PGL(3,q) and PGammaL(3,q).

Cf. A373501 for the size of the collineation groups.

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

Map[(#^2+#-1)!/(PrimeOmega[#]*(#-1)^2*#^2) &, Select[Range[10], PrimePowerQ]] (* Paolo Xausa, Aug 01 2024 *)

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

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

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User: Ralf Goertz

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

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