Projective general semilinear groups

The projective general semilinear group PΓL(n,𝔽q) is ΓL(n,𝔽_q) modulo the scalars {λI} ≅ Cq−1 — its faithful action on the projective space PG(n−1, q):

PΓL(n,𝔽q) = PGL(n,𝔽q) ⋊ Gal(𝔽q / 𝔽p) ≅ PGL(n,𝔽q) ⋊ Cd, where q = pd.

• Order — |PΓL(n,𝔽q)| = d · |PGL(n,𝔽q)| = d · |GL(n,𝔽q)| / (q − 1).
• Over a prime field — PΓL(n,𝔽p) = PGL(n,𝔽p).
• Collineations — for n ≥ 3 the fundamental theorem of projective geometry says PΓL(n,𝔽q) is the full automorphism group of PG(n−1, q) as an incidence geometry: every collineation is semilinear. (For n = 2 there is no incidence structure to preserve — any three points of a line are collinear — so the statement starts at projective dimension 2.)

Automorphisms of PSL

PΓL(n,𝔽q) supplies the diagonal and field automorphisms of the simple group PSL(n,𝔽_q):

• n = 2, q > 3 — Aut(PSL(2,𝔽q)) = PΓL(2,𝔽q).
• n ≥ 3 — Aut(PSL(n,𝔽q)) = PΓL(n,𝔽q) ⋊ C2, the extra factor being the graph automorphism A ↦ (Aᵀ)−1 (transpose-inverse), which PΓL does not contain.

Small cases

• PΓL(2,𝔽4) ≅ S₅ (order 120) — PSL(2,𝔽4) ≅ A₅ extended by the Frobenius of 𝔽4.
• PΓL(2,𝔽8) — order 3 · 504 = 1512, the full automorphism group of the simple group PSL(2,𝔽8).
• PΓL(2,𝔽9) — order 2 · 720 = 1440, equal to Aut(A₆) since PSL(2,𝔽9) ≅ A₆. Its three subgroups of index 2 over A₆ are PGL(2,𝔽9), PΣL(2,𝔽₉) ≅ S₆, and the Mathieu group M₁₀.
• PΓL(2,𝔽16) — order 4 · 4080 = 16 320.
• PΓL(2,𝔽25) — order 2 · 15 600 = 31 200.

Part of the wider family of matrix groups: the projective quotient of ΓL, containing PΣL with quotient Cgcd(n, q−1).

Wikipedia: Projective linear group