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

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

Offset: 1

Author

Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001

Keywords

• nonn

Comments

From Jianing Song, Nov 06 2019: (Start)

GL_2(K) means the group of invertible 2 X 2 matrices A over K.

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)

Examples

			a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• R. A. Wilson, The classical groups, chapter 3.3.1 in The finite Simple Groups, Graduate Texts in Mathematics 251 (2009).

Crossrefs

Subsequence of A047927.

Cf. A246655, A000252 (order of GL_2(Z_n)).

For the order of SL_2(K) see A329119.

Programs

Maple

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:

Mathematica

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 *)

PARI

[(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ Jianing Song, Nov 05 2019

Formula

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.

a(n) = A047927(A246655(n)+1). - Jianing Song, Nov 05 2019

a(n) = (A246655(n)-1)*A329119(n). - Jianing Song, Nov 06 2019

Extensions

More terms from James Sellers, Jan 22 2001

Offset corrected by Jianing Song, Nov 05 2019