Projective special linear groups

The projective special linear group PSL(n,𝔽q) is SL(n,𝔽q) modulo its center {λI : λⁿ = 1} — the action of SL on the projective space PG(n−1, q).

• Order — |PSL(n,𝔽q)| = |SL(n,𝔽q)| / gcd(n, q − 1); for n = 2, q(q² − 1) / gcd(2, q − 1).
• Simple — for every n ≥ 2 and prime power q, except PSL(2,𝔽2) ≅ S₃ and PSL(2,𝔽3) ≅ A₄. Together with the cyclic Cp and the alternating Aₙ, the classical infinite families of finite simple groups (Lie type Aₙ₋₁).
• Action — 2-transitive on the q + 1 points of the projective line (n = 2); the sharply 3-transitive closure is PGL(2,𝔽_q).

Exceptional isomorphisms

The small members collide with the alternating family:

• PSL(2,𝔽4) ≅ PSL(2,𝔽5) ≅ A₅ — order 60, the smallest non-abelian simple group.
• PSL(2,𝔽7) ≅ PSL(3,𝔽2) — order 168, the second smallest.
• PSL(2,𝔽9) ≅ A₆ — order 360.
• PSL(4,𝔽2) ≅ A₈ — order 20 160.

Part of the wider family of matrix groups: the projective quotient of SL, normal in PGL with cyclic quotient of order gcd(n, q − 1).

Wikipedia: Projective linear group