OFFSET
1,1
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)
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).
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.
EXAMPLE
a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.
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 *)
PROG
(PARI) [(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ Jianing Song, Nov 05 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001
EXTENSIONS
More terms from James A. Sellers, Jan 22 2001
Offset corrected by Jianing Song, Nov 05 2019
STATUS
approved