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

%I A373501 #29 Aug 02 2024 14:12:22
%S A373501 168,5616,120960,372000,5630688,49448448,84913920,212427600,810534816,
%T A373501 17108582400,6950204928,16934047920,78156525216,304668000000,
%U A373501 846083360304,499631102880,851974934400,5492021821440,3509844434208,7980059337600,11681731985616,23800278205248
%N A373501 Size of the collineation group of classical projective planes of prime power order q.
%C A373501 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.
%D A373501 A. Beutelspacher and U. Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998, pages 118-132.
%D A373501 D. R. Hughes and F. C. Piper, Projective Planes, Springer, 1973.
%H A373501 Paolo Xausa, <a href="/A373501/b373501.txt">Table of n, a(n) for n = 1..10000</a>
%H A373501 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_linear_group">Projective linear group</a>, order of PGL(3,q) and PGammaL(3,q).
%F A373501 a(n) = Omega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) where q = A246655(n).
%e A373501 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:
%e A373501             (0 1 2 3 4 5 6)   (0 6)(3 5)
%e A373501   {0,1,3}       {1,2,4}        {6,1,5}
%e A373501   {1,2,4}       {2,3,5}        {1,2,4}
%e A373501   {2,3,5}       {3,4,6}        {2,5,3}
%e A373501   {3,4,6}       {4,5,0}        {5,4,0}
%e A373501   {4,5,0}       {5,6,1}        {4,3,6}
%e A373501   {5,6,1}       {6,0,2}        {3,0,1}
%e A373501   {6,0,2}       {0,1,3}        {0,6,2}
%e A373501 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.
%t A373501 Map[PrimeOmega[#]*#^3*(#^2+#+1)*(#^2-1)*(#-1) &, Select[Range[50], PrimePowerQ]] (* _Paolo Xausa_, Aug 01 2024 *)
%o A373501 (PARI) a=(q)->bigomega(q)*(q^3-1)*(q^3-q)*(q^3-q^2)/(q-1) \\ q=A246655(n)
%Y A373501 Cf. A373502 for the size of a complete set of classical projective planes using a given set of q^2+q+1 points.
%Y A373501 Cf. A335866 for the number of projective planes whose lines are cyclic difference sets.
%K A373501 nonn
%O A373501 1,1
%A A373501 _Ralf Goertz_, Jun 07 2024
%E A373501 Data corrected by _Paolo Xausa_, Aug 02 2024