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A329119
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Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.
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5
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6, 24, 60, 120, 336, 504, 720, 1320, 2184, 4080, 4896, 6840, 12144, 15600, 19656, 24360, 29760, 32736, 50616, 68880, 79464, 103776, 117600, 148824, 205320, 226920, 262080, 300696, 357840, 388944, 492960, 531360, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920
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OFFSET
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1,1
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COMMENTS
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SL_2(K) means the group of 2 X 2 matrices A over K such that det(A) = 1.
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.
Also a(n) is the order of PGL_2(K) when K is a finite field with q = A246655(n) elements. Note that PGL(m,q) and SL(m,q) are not isomorphic unless gcd(m,q-1) = 1. For example, PGL(2,3) = S_4 is not isomorphic to SL(2,3), PGL(2,5) = S_5 is not isomorphic to SL(2,5). - Jianing Song, Apr 04 2022
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LINKS
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FORMULA
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If the finite field K has q elements, then the order of the group SL_2(K) is q*(q^2-1).
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EXAMPLE
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a(4) = 120 because A246655(4) = 5, and 5*(5^2-1) = 120.
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MAPLE
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N:= 200:
P:= select(isprime, {2, seq(i, i=3..N, 2)}):
PP:= map(proc(p) local i; seq(p^i, i=1..floor(log[p](N))) end proc, P):
map(t -> t*(t^2-1), sort(convert(PP, list))); # Robert Israel, Nov 13 2019
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MATHEMATICA
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p = Select[Range[200], PrimePowerQ];
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PROG
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(PARI) [(p+1)*p*(p-1) | p <- [1..200], isprimepower(p)]
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CROSSREFS
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For the order of GL_2(K) see A059238.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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