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A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements. 9
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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

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

Subsequence of A047927.

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

For the order of SL_2(K) see A329119.

Sequence in context: A331668 A005353 A047927 * A254832 A026695 A208536

Adjacent sequences:  A059235 A059236 A059237 * A059239 A059240 A059241

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

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Last modified August 13 18:03 EDT 2022. Contains 356107 sequences. (Running on oeis4.)