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%I #21 Jun 05 2020 15:16:00
%S 6,48,168,180,480,2016,3528,5760,11232,13200,20160,26208,61200,78336,
%T 123120,181440,267168,374400,511056,682080,892800,1014816,1488000,
%U 1822176,2755200,3337488,4773696,5644800,7738848,9999360,11908560,13615200,16511040,19845936,24261120,25048800,28003968
%N Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.
%C 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.
%C 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.
%C Some isomorphisms (let "==" denote "isomorphic to"):
%C a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
%C a(2) = 48 for GL(2,3) that has 55 subgroups.
%C a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
%C a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
%C Array for order of GL(m,q) begins:
%C =============================================================
%C m\q | 2 3 4=2^2 5 7
%C -------------------------------------------------------------
%C 2 | 6 48 180 480 2016
%C 3 | 168 11232 181440 1488000 33784128
%C 4 | 20160 24261120 2961100800 116064000000 #GL(4,7)
%C 5 |9999360 #GL(5,3) ... ... ...
%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%D Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/General_linear_group">General linear group</a>
%F #GL(m,q) = Product_{k=0..m-1}(q^m-q^k).
%e 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:
%e (1 0) (1 1) (1 0) (0 1) (0 1) (1 1)
%e (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
%e a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
%e a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.
%Y Cf. A059238 [GL(2,q)].
%Y 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)].
%Y Cf. A316622 [GL(n,Z_k)].
%K nonn
%O 1,1
%A _Bernard Schott_, Jun 04 2020
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