Projective general linear groups

The projective general linear group PGL(n,𝔽q) is GL(n,𝔽q) modulo its center, the scalar matrices {λI} ≅ Cq−1 — the faithful action of GL on the projective space PG(n−1, q), whose points are the (qⁿ − 1)/(q − 1) lines through 0 of 𝔽qn.

• Order — |PGL(n,𝔽q)| = |GL(n,𝔽q)| / (q − 1) — the same order as SL(n,𝔽_q), though the groups differ in general: both PGL(2,𝔽3) ≅ S₄ and SL(2,𝔽3) have order 24, but SL(2,𝔽3) has a unique involution −I while S₄ has nine.
• PSL inside — PSL(n,𝔽_q) is normal in PGL(n,𝔽q) with cyclic quotient of order gcd(n, q − 1), so PGL = PSL exactly when gcd(n, q − 1) = 1.

The projective line

PGL(2,𝔽q) acts on the q + 1 points of the projective line as the Möbius (fractional linear) transformations x ↦ (ax + b)/(cx + d), ad − bc ≠ 0. The action is sharply 3-transitive: exactly one element carries any ordered triple of distinct points to any other — matching |PGL(2,𝔽q)| = (q + 1) q (q − 1).

Small cases

• PGL(2,𝔽2) = PSL(2,𝔽2) ≅ S₃, all six permutations of the 3 points of PG(1, 2).
• PGL(2,𝔽3) ≅ S₄ — all permutations of the 4 points of PG(1, 3).
• PGL(2,𝔽4) = PSL(2,𝔽4) ≅ A₅ (order 60).
• PGL(2,𝔽5) ≅ S₅ (order 120), acting 3-transitively on the 6 points of PG(1, 5) — the exceptional transitive embedding S₅ ↪ S₆ behind the outer automorphism of S₆.
• PGL(3,𝔽2) = PSL(3,𝔽2) ≅ PSL(2,𝔽₇), simple of order 168.

Part of the wider family of matrix groups: the projective quotient of GL, containing PSL.

Wikipedia: Projective linear group