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

6, 48, 240, 480, 1344, 9072, 11520, 10560, 17472, 130560, 78336, 82080, 242880, 499200, 1415232, 584640, 476160, 4910400, 1214784, 2204160, 1907136, 4566144, 7526400, 7143552, 11497920, 7261440, 56609280, 12027840, 17176320, 18669312, 23662080, 136028160, 45736320, 56390400, 58404864, 82416000, 69927936

Offset

1,1

Comments

For orders of Aut(SL_2(K)) = Aut(PGL_2(K)) = Aut(PSL_2(K)) see A352807.

See the Groupprops link for a formula for |Aut(GL(n,q))| in general.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Groupprops, Endomorphism structure of general linear group over a finite field

Formula

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).

Example

a(5) = 1344 since A246655(5) = 7, so a(5) = A352807(5)*eulerphi(2*(7-1)) = 336*4 = 1344.
a(6) = 9072 since A246655(6) = 8, so a(6) = A352807(6)*eulerphi(2*(8-1)) = 1512*6 = 9072.
a(7) = 11520 since A246655(7) = 9, so a(7) = A352807(7)*eulerphi(2*(9-1)) = 1440*8 = 15120.

Prog

(PARI) [(q+1)*q*(q-1)*isprimepower(q)*eulerphi(2*(q-1)) | q <- [1..200], isprimepower(q)]

Crossrefs

Cf. A246655.

Order of GL(2,q): A059238;

SL(2,q): A329119;

PGL(2,q): A329119;

PSL(2,q): A352806;

Aut(GL(2,q)): this sequence;

PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.

Sequence in context: A208536 A253947 A260344 * A262354 A052771 A056289

Adjacent sequences: A353244 A353245 A353246 * A353248 A353249 A353250

Keyword

nonn

Author

Jianing Song, Apr 08 2022

Status

approved