A352807 - OEIS

%I #18 Aug 12 2022 19:00:10

%S 6,24,120,120,336,1512,1440,1320,2184,16320,4896,6840,12144,31200,

%T 58968,24360,29760,163680,50616,68880,79464,103776,235200,148824,

%U 205320,226920,1572480,300696,357840,388944,492960,2125440,571704,704880,912576,1030200,1092624

%N Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

%C PGammaL_n(K) is the projective semilinear group of order n over K (see Wikipedia link). It is the semidirect product of PGL_n(K) and Aut(K), where Aut(K) is the group of field automorphisms of K. So if p is a prime, then PGammaL(n,p) is isomorphic to PGL(n,p).

%C We also have Aut(SL_n(K)) = Aut(PGL_n(K)) = Aut(PSL_n(K)) for arbitrary field K, and when n = 2 this is isomorphic to PGammaL_2(K). If n >= 3, this is isomorphic to the semidirect product of PGammaL_2(K) and C_2.

%C Examples are PGammaL(2,2) = S_3, PGammaL(2,3) = S_4, PGammaL(2,4) = PGammaL(2,5) = S_5, PGammaL(2,9) = Aut(S_6) = Aut(A_6).

%H Jianing Song, <a href="/A352807/b352807.txt">Table of n, a(n) for n = 1..10000</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Projective_semilinear_group">Projective semilinear group</a>

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4419869/do-the-groups-operatornamesl-operatornamepgl-and-operatornamepsl">Do the groups SL, PGL, and PSL over a field K always have the same automorphism group?</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Semilinear_map#projective_semilinear_group">Semilinear map</a>

%F For q = p^r, |PGammaL(2,q)| = r*q*(q^2-1) = r*|PGL(2,q)|. In general, |PGammaL(n,q)| = r*|PGL(n,q)|.

%e a(6) = 1512 since A246655(6) = 8 = 2^3, so a(6) = 3*A329119(6) = 3*504 = 1512.

%e a(7) = 1440 since A246655(7) = 9 = 3^2, so a(7) = 2*A329119(7) = 2*720 = 1440.

%o (PARI) [(q+1)*q*(q-1)*isprimepower(q) | q <- [1..200], isprimepower(q)]

%Y Cf. A246655.

%Y Order of GL(2,q): A059238;

%Y SL(2,q): A329119;

%Y PGL(2,q): A329119;

%Y PSL(2,q): A352806;

%Y Aut(GL(2,q)): A353247;

%Y PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): this sequence.

%K nonn

%O 1,1

%A _Jianing Song_, Apr 04 2022