OFFSET
1,1
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
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
FORMULA
EXAMPLE
a(4) = 120 because A246655(4) = 5, and 5*(5^2-1) = 120.
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 *)
PROG
(PARI) [(p+1)*p*(p-1) | p <- [1..200], isprimepower(p)]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Nov 05 2019
STATUS
approved