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A059238
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Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.
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9
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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
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OFFSET
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1,1
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COMMENTS
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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)
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LINKS
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R. A. Wilson, The classical groups, chapter 3.3.1 in The finite Simple Groups, Graduate Texts in Mathematics 251 (2009).
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FORMULA
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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.
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EXAMPLE
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a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(PARI) [(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ Jianing Song, Nov 05 2019
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CROSSREFS
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For the order of SL_2(K) see A329119.
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KEYWORD
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nonn
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001
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EXTENSIONS
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STATUS
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approved
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