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).
Order of the automorphism group of the field F_q(T), q = A246655(n). Note that when we view PGammaL_2(K) as Aut(K(T)), it is the semidirect product of Aut(K(T)/K) (isomorphic to PGL_2(K)) and Aut(K(T)/k(T)) (isomorphic to Aut(K)), where k is the prime field of K. Here, PGammaL(2,q) = Aut(F_q(T)) is the semidirect product of PGL(2,q) = Aut(F_q(T)/F_q) and Aut(F_q) = Gal(F_q(T)/F_p(T)). - Jianing Song, Oct 09 2025
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
MATHEMATICA
Map[(#-1) * # * (#+1) * FactorInteger[#][[1, 2]] &, Select[Range[120], PrimePowerQ]] (* Amiram Eldar, Mar 28 2026 *)
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
