OFFSET
1,1
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).
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
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)|.
EXAMPLE
PROG
(PARI) [(q+1)*q*(q-1)*isprimepower(q) | q <- [1..200], isprimepower(q)]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Apr 04 2022
STATUS
approved