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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)) = 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.)