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.

%I A335384 #22 Jun 14 2024 22:31:11
%S A335384 6,48,168,180,480,2016,3528,5760,11232,13200,20160,26208,61200,78336,
%T A335384 123120,181440,267168,374400,511056,682080,892800,1014816,1488000,
%U A335384 1822176,2755200,3337488,4773696,5644800,7738848,9999360,11908560,13615200,16511040,19845936,24261120,25048800,28003968
%N A335384 Order of the finite groups GL(m,q) [or GL_m(q)] in increasing order as q runs through the prime powers.
%C A335384 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 A335384 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 A335384 Some isomorphisms (let "==" denote "isomorphic to"):
%C A335384 a(1) = 6 for GL(2,2) == PSL(2,2) == S_3.
%C A335384 a(2) = 48 for GL(2,3) that has 55 subgroups.
%C A335384 a(3) = 168 for GL(3,2) == PSL(2,7) [A031963].
%C A335384 a(11) = 20160 for GL(4,2) == PSL(4,2) == Alt(8).
%C A335384 Array for order of GL(m,q) begins:
%C A335384 =============================================================
%C A335384 m\q  |      2         3       4=2^2             5         7
%C A335384 -------------------------------------------------------------
%C A335384   2  |      6        48         180           480      2016
%C A335384   3  |    168     11232      181440       1488000  33784128
%C A335384   4  |  20160  24261120  2961100800  116064000000   #GL(4,7)
%C A335384   5  |9999360  #GL(5,3)      ...           ...         ...
%D A335384 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].
%D A335384 Daniel Perrin, Cours d'Algèbre, Maths Agreg, Ellipses, 1996, pages 95-115.
%H A335384 Wikipedia, <a href="https://en.wikipedia.org/wiki/General_linear_group">General linear group</a>
%F A335384 #GL(m,q) = Product_{k=0..m-1}(q^m-q^k).
%e A335384 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 A335384   (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
%e A335384   (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
%e A335384 a(2) = #GL(2,3) = (3^2-1)*(3^2-3) = 8*6 = 48.
%e A335384 a(3) = #GL(3,2) = (2^3-1)*(2^3-2)*(2^3-2^2) = 168.
%Y A335384 Cf. A059238 [GL(2,q)].
%Y A335384 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 A335384 Cf. A316622 [GL(n,Z_k)].
%K A335384 nonn
%O A335384 1,1
%A A335384 _Bernard Schott_, Jun 04 2020