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

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

Author

Jianing Song, Apr 04 2022

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• 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

Crossrefs

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.

Programs

PARI

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

Formula

For q = p^r, |PGammaL(2,q)| = r*q*(q^2-1) = r*|PGL(2,q)|. In general, |PGammaL(n,q)| = r*|PGL(n,q)|.