A329119 - OEIS

%I #37 Apr 04 2022 10:21:04

%S 6,24,60,120,336,504,720,1320,2184,4080,4896,6840,12144,15600,19656,

%T 24360,29760,32736,50616,68880,79464,103776,117600,148824,205320,

%U 226920,262080,300696,357840,388944,492960,531360,571704,704880,912576,1030200,1092624,1224936,1294920

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

%C SL_2(K) means the group of 2 X 2 matrices A over K such that det(A) = 1.

%C 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.

%C 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

%H Robert Israel, <a href="/A329119/b329119.txt">Table of n, a(n) for n = 1..10000</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Projective_general_linear_group_of_degree_two">Projective general linear group of degree two</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Special_linear_group_of_degree_two">Special linear group of degree two</a>

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

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

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

%p N:= 200:

%p P:= select(isprime, {2,seq(i,i=3..N,2)}):

%p PP:= map(proc(p) local i; seq(p^i,i=1..floor(log[p](N))) end proc, P):

%p map(t -> t*(t^2-1), sort(convert(PP,list))); # _Robert Israel_, Nov 13 2019

%t p = Select[Range[200], PrimePowerQ];

%t (p-1) p (p+1) (* _Jean-François Alcover_, Aug 22 2020 *)

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

%Y Subsequence of A007531.

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

%Y For the order of GL_2(K) see A059238.

%K nonn

%O 1,1

%A _Jianing Song_, Nov 05 2019