cp's OEIS Frontend

A353247 Orders of the finite groups Aut(GL_2(K)) when K is a finite field with q = A246655(n) elements.

%I A353247 #12 Apr 09 2022 09:47:31
%S A353247 6,48,240,480,1344,9072,11520,10560,17472,130560,78336,82080,242880,
%T A353247 499200,1415232,584640,476160,4910400,1214784,2204160,1907136,4566144,
%U A353247 7526400,7143552,11497920,7261440,56609280,12027840,17176320,18669312,23662080,136028160,45736320,56390400,58404864,82416000,69927936
%N A353247 Orders of the finite groups Aut(GL_2(K)) when K is a finite field with q = A246655(n) elements.
%C A353247 For orders of Aut(SL_2(K)) = Aut(PGL_2(K)) = Aut(PSL_2(K)) see A352807.
%C A353247 See the Groupprops link for a formula for |Aut(GL(n,q))| in general.
%H A353247 Jianing Song, <a href="/A353247/b353247.txt">Table of n, a(n) for n = 1..10000</a>
%H A353247 Groupprops, <a href="https://groupprops.subwiki.org/wiki/Endomorphism_structure_of_general_linear_group_over_a_finite_field">Endomorphism structure of general linear group over a finite field</a>
%F A353247 For q = p^r, |Aut(GL(2,q))| = r*q*(q^2-1)*eulerphi(2*(q-1)) = |PGammaL(2,q)|*eulerphi(2*(q-1)) (see A352807). In general, we have |Aut(GL(n,q))|/|Aut(SL(n,q))| = eulerphi(n*(q-1))/eulerphi(n).
%e A353247 a(5) = 1344 since A246655(5) = 7, so a(5) = A352807(5)*eulerphi(2*(7-1)) = 336*4 = 1344.
%e A353247 a(6) = 9072 since A246655(6) = 8, so a(6) = A352807(6)*eulerphi(2*(8-1)) = 1512*6 = 9072.
%e A353247 a(7) = 11520 since A246655(7) = 9, so a(7) = A352807(7)*eulerphi(2*(9-1)) = 1440*8 = 15120.
%o A353247 (PARI) [(q+1)*q*(q-1)*isprimepower(q)*eulerphi(2*(q-1)) | q <- [1..200], isprimepower(q)]
%Y A353247 Cf. A246655.
%Y A353247 Order of GL(2,q): A059238;
%Y A353247 SL(2,q): A329119;
%Y A353247 PGL(2,q): A329119;
%Y A353247 PSL(2,q): A352806;
%Y A353247 Aut(GL(2,q)): this sequence;
%Y A353247 PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.
%K A353247 nonn
%O A353247 1,1
%A A353247 _Jianing Song_, Apr 08 2022