cp's OEIS Frontend

A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

%I A352807 #18 Aug 12 2022 19:00:10
%S A352807 6,24,120,120,336,1512,1440,1320,2184,16320,4896,6840,12144,31200,
%T A352807 58968,24360,29760,163680,50616,68880,79464,103776,235200,148824,
%U A352807 205320,226920,1572480,300696,357840,388944,492960,2125440,571704,704880,912576,1030200,1092624
%N A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.
%C A352807 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 A352807 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 A352807 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 A352807 Jianing Song, <a href="/A352807/b352807.txt">Table of n, a(n) for n = 1..10000</a>
%H A352807 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Projective_semilinear_group">Projective semilinear group</a>
%H A352807 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 A352807 Wikipedia, <a href="https://en.wikipedia.org/wiki/Semilinear_map#projective_semilinear_group">Semilinear map</a>
%F A352807 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 A352807 a(6) = 1512 since A246655(6) = 8 = 2^3, so a(6) = 3*A329119(6) = 3*504 = 1512.
%e A352807 a(7) = 1440 since A246655(7) = 9 = 3^2, so a(7) = 2*A329119(7) = 2*720 = 1440.
%o A352807 (PARI) [(q+1)*q*(q-1)*isprimepower(q) | q <- [1..200], isprimepower(q)]
%Y A352807 Cf. A246655.
%Y A352807 Order of GL(2,q): A059238;
%Y A352807 SL(2,q): A329119;
%Y A352807 PGL(2,q): A329119;
%Y A352807 PSL(2,q): A352806;
%Y A352807 Aut(GL(2,q)): A353247;
%Y A352807 PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): this sequence.
%K A352807 nonn
%O A352807 1,1
%A A352807 _Jianing Song_, Apr 04 2022