A352806 - OEIS

%I #19 Jul 03 2023 05:53:52

%S 6,12,60,60,168,504,360,660,1092,4080,2448,3420,6072,7800,9828,12180,

%T 14880,32736,25308,34440,39732,51888,58800,74412,102660,113460,262080,

%U 150348,178920,194472,246480,265680,285852,352440,456288,515100,546312,612468,647460

%N Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

%C For a communtative unital ring R, PSL_n(R), the projective special linear group of order n over R, is defined as SL_n(R)/{r*I_n: r^n = 1}. This is related to PGL_n(R), the projective general linear group of order n over R, which is defined as GL_n(R)/{r*I_n: r is a unit of R}.

%C Note that a(3) = a(4) = 60 refer to the same group (PSL(2,4) = PSL(2,5) = Alt(5)). Also PSL(2,9) = Alt(6).

%H Jianing Song, <a href="/A352806/b352806.txt">Table of n, a(n) for n = 1..10000</a>

%H Groupprops, <a href="https://groupprops.subwiki.org/wiki/Projective_special_linear_group">Projective special linear group</a>

%F |PSL(2,q)| = q*(q^2-1)/2 if q is odd, q*(q^2-1) otherwise.

%F |PSL(2,q)| = |PGL(2,q)|/gcd(2,q-1) = |SL(2,q)|/gcd(2,q-1).

%F In general, |PSL(n,q)| = |PGL(n,q)|/gcd(n,q-1) = |SL(n,q)|/gcd(n,q-1).

%e a(6) = 504 since A246655(6) = 8, so a(6) = 8*(8^2-1)/gcd(2,8-1) = 504.

%e a(7) = 360 since A246655(7) = 9, so a(7) = 9*(9^2-1)/gcd(2,9-1) = 360.

%o (PARI) [(q+1)*q*(q-1)/gcd(2,q-1) | q <- [1..200], isprimepower(q)]

%Y Cf. A246655.

%Y Order of GL(2,q): A059238;

%Y SL(2,q): A329119;

%Y PGL(2,q): A329119;

%Y PSL(2,q): this sequence;

%Y Aut(GL(2,q)): A353247;

%Y PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.

%Y A117762 is a subsequence, A335000 is a supersequence.

%K nonn

%O 1,1

%A _Jianing Song_, Apr 04 2022