A316622 -id:A316622 - cp's OEIS frontend

A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

6, 48, 168, 180, 480, 2016, 3528, 5760, 11232, 13200, 20160, 26208, 61200, 78336, 123120, 181440, 267168, 374400, 511056, 682080, 892800, 1014816, 1488000, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 9999360, 11908560, 13615200, 16511040, 19845936, 24261120, 25048800, 28003968

Offset: 1

Bernard Schott, Jun 04 2020

nonn

GL(m,q) is the general linear group, the group of invertible m X m matrices over the finite field F_q with q = p^k elements.
By definition, all fields must contain at least two distinct elements, so q >= 2. As GL(1,q) is isomorphic to F_q*, the multiplicative group [whose order is p^k-1 (A181062)] of finite field F_q, data begins with m >= 2.
Some isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
a(2) = 48 for GL(2,3) that has 55 subgroups.
a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
Array for order of GL(m,q) begins:
=============================================================
m\q  |      2         3       4=2^2             5         7
-------------------------------------------------------------
  2  |      6        48         180           480      2016
  3  |    168     11232      181440       1488000  33784128
  4  |  20160  24261120  2961100800  116064000000   #GL(4,7)
  5  |9999360  #GL(5,3)      ...           ...         ...

			a(1) = #GL(2,2) = (2^2-1)*(2^2-2) = 3*2 = 6 and the 6 elements of GL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 invertible matrices with entries in F_2:
  (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
  (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.

Wikipedia, General linear group

Cf. A059238 [GL(2,q)].
Cf. A002884 [GL(m,2)], A053290 [GL(m,3)], A053291 [GL(m,4)], A053292 [GL(m,5)], A053293 [GL(m,7)], A052496 [GL(m,8)], A052497 [GL(m,9)], A052498 [GL(m,11)].
Cf. A316622 [GL(n,Z_k)].

#GL(m,q) = Product_{k=0..m-1}(q^m-q^k).

cp's OEIS Frontend

A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.

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