A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

6, 12, 60, 60, 168, 504, 360, 660, 1092, 4080, 2448, 3420, 6072, 7800, 9828, 12180, 14880, 32736, 25308, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 262080, 150348, 178920, 194472, 246480, 265680, 285852, 352440, 456288, 515100, 546312, 612468, 647460

Offset: 1

Author

Jianing Song, Apr 04 2022

Keywords

• nonn

Comments

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}.

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).

Examples

			a(6) = 504 since A246655(6) = 8, so a(6) = 8*(8^2-1)/gcd(2,8-1) = 504.
a(7) = 360 since A246655(7) = 9, so a(7) = 9*(9^2-1)/gcd(2,9-1) = 360.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Groupprops, Projective special linear group

Crossrefs

Cf. A246655.

Order of GL(2,q): A059238;

SL(2,q): A329119;

PGL(2,q): A329119;

PSL(2,q): this sequence;

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

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

A117762 is a subsequence, A335000 is a supersequence.

Programs

PARI

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

Formula

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

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

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