cp's OEIS Frontend

A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.

%I A059238 #38 Jul 02 2025 16:02:00
%S A059238 6,48,180,480,2016,3528,5760,13200,26208,61200,78336,123120,267168,
%T A059238 374400,511056,682080,892800,1014816,1822176,2755200,3337488,4773696,
%U A059238 5644800,7738848,11908560,13615200,16511040,19845936,25048800,28003968
%N A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.
%C A059238 From _Jianing Song_, Nov 06 2019: (Start)
%C A059238 GL_2(K) means the group of invertible 2 X 2 matrices A over K.
%C A059238 In general, let R be any commutative ring with unity, GL_n(R) be the group of n X n matrices A over R such that det(A) != 0 and SL_n(R) be the group of n X n matrices A over R such that det(A) = 1, then GL_n(R)/SL_n(R) = R* is the multiplicative group of R. This is because if we define f(M) = det(M) for M in GL_n(R), then f is a surjective homomorphism from GL_n(K) to R*, and SL_n(R) is its kernel. Thus |GL_n(R)|/|SL_n(R)| = |R*|; if K is a finite field, then |GL_n(R)|/|SL_n(R)| = |K|-1. (End)
%H A059238 Jianing Song, <a href="/A059238/b059238.txt">Table of n, a(n) for n = 1..10000</a>
%H A059238 R. A. Wilson, <a href="http://dx.doi.org/10.1007/978-1-84800-988-2_3">The classical groups</a>, chapter 3.3.1 in The finite Simple Groups, Graduate Texts in Mathematics 251 (2009).
%F A059238 If the finite field K has p^m elements, then the order of the group GL_2(K) is (p^(2m)-1)*(p^(2m)-p^m) = (p^m+1)*(p^m)*(p^m-1)^2.
%F A059238 a(n) = A047927(A246655(n)+1). - _Jianing Song_, Nov 05 2019
%F A059238 a(n) = (A246655(n)-1)*A329119(n). - _Jianing Song_, Nov 06 2019
%e A059238 a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.
%p A059238 with(numtheory): for n from 2 to 400 do if nops(ifactors(n)[2]) = 1 then printf(`%d,`, (n+1)*(n)*(n-1)^2) fi: od:
%t A059238 nn=30;a=Take[Union[Sort[Flatten[Table[Table[Prime[m]^k,{m,1,nn}],{k,1,nn}]]]],nn];Table[(q^2-1)(q^2-q),{q,a}]  (* _Geoffrey Critzer_, Apr 20 2013 *)
%o A059238 (PARI) [(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ _Jianing Song_, Nov 05 2019
%Y A059238 Subsequence of A047927.
%Y A059238 Cf. A246655, A000252 (order of GL_2(Z_n)).
%Y A059238 For the order of SL_2(K) see A329119.
%K A059238 nonn
%O A059238 1,1
%A A059238 Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001
%E A059238 More terms from _James Sellers_, Jan 22 2001
%E A059238 Offset corrected by _Jianing Song_, Nov 05 2019