A329119 Orders of the finite groups SL_2(K) when K is a finite field with q = A246655(n) elements.

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

Offset: 1

Author

Jianing Song, Nov 05 2019

Keywords

• nonn

Comments

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

Examples

			a(4) = 120 because A246655(4) = 5, and 5*(5^2-1) = 120.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Groupprops, Projective general linear group of degree two
• Groupprops, Special linear group of degree two

Crossrefs

Subsequence of A007531.

Cf. A246655, A000056 (order of SL_2(Z_n)).

For the order of GL_2(K) see A059238.

Programs

Maple

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

Mathematica

p = Select[Range[200], PrimePowerQ];
(p-1) p (p+1) (* Jean-François Alcover, Aug 22 2020 *)

PARI

[(p+1)*p*(p-1) | p <- [1..200], isprimepower(p)]

Formula

If the finite field K has q elements, then the order of the group SL_2(K) is q*(q^2-1).

a(n) = A059238(n)/(A246655(n)-1) = A007531(A246655(n)+1).