A353247 -id:A353247 - cp's OEIS frontend

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

Jianing Song, Apr 04 2022

nonn

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).

			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.

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

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.

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

|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).

A352807 Orders of the finite groups PGammaL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 24, 120, 120, 336, 1512, 1440, 1320, 2184, 16320, 4896, 6840, 12144, 31200, 58968, 24360, 29760, 163680, 50616, 68880, 79464, 103776, 235200, 148824, 205320, 226920, 1572480, 300696, 357840, 388944, 492960, 2125440, 571704, 704880, 912576, 1030200, 1092624

Offset: 1

Jianing Song, Apr 04 2022

nonn

PGammaL_n(K) is the projective semilinear group of order n over K (see Wikipedia link). It is the semidirect product of PGL_n(K) and Aut(K), where Aut(K) is the group of field automorphisms of K. So if p is a prime, then PGammaL(n,p) is isomorphic to PGL(n,p).
We also have Aut(SL_n(K)) = Aut(PGL_n(K)) = Aut(PSL_n(K)) for arbitrary field K, and when n = 2 this is isomorphic to PGammaL_2(K). If n >= 3, this is isomorphic to the semidirect product of PGammaL_2(K) and C_2.
Examples are PGammaL(2,2) = S_3, PGammaL(2,3) = S_4, PGammaL(2,4) = PGammaL(2,5) = S_5, PGammaL(2,9) = Aut(S_6) = Aut(A_6).

			a(6) = 1512 since A246655(6) = 8 = 2^3, so a(6) = 3*A329119(6) = 3*504 = 1512.
a(7) = 1440 since A246655(7) = 9 = 3^2, so a(7) = 2*A329119(7) = 2*720 = 1440.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective semilinear group
Mathematics Stack Exchange, Do the groups SL, PGL, and PSL over a field K always have the same automorphism group?
Wikipedia, Semilinear map

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): A352806;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): this sequence.

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

cp's OEIS Frontend

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

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