Projective special semilinear groups

The projective special semilinear group PΣL(n,𝔽q) is ΣL(n,𝔽_q) modulo the scalars it contains — the image of the determinant-1 semilinear maps in the projective group:

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

• Order — |PΣL(n,𝔽q)| = d · |PSL(n,𝔽q)|. The kernel of the projective action on ΣL is {λI : λⁿ = 1}, of order gcd(n, q − 1), exactly as for SL.
• Over a prime field — PΣL(n,𝔽p) = PSL(n,𝔽p).
• Inside PΓL — normal, with PΓL / PΣL ≅ PGL / PSL ≅ Cgcd(n, q−1); the two are equal exactly when gcd(n, q − 1) = 1 — e.g. always in dimension 2 over an even field.
• Simple socle — PSL(n,𝔽q) is simple (bar PSL(2,𝔽2) and PSL(2,𝔽3)), so PΣL(n,𝔽q) is an almost simple group: the field automorphisms sitting on top of a simple group.

Small cases

• PΣL(2,𝔽4) = PΓL(2,𝔽4) ≅ S₅ (order 120): gcd(2, 3) = 1 collapses the two names.
• PΣL(2,𝔽8) — order 3 · 504 = 1512, again equal to PΓL(2,𝔽8).
• PΣL(2,𝔽9) ≅ S₆ (order 720). Since PSL(2,𝔽9) ≅ A₆, this is A₆ extended by the Frobenius of 𝔽9 — one of the three index-2 subgroups of PΓL(2,𝔽₉) over A₆, the others being PGL(2,𝔽9) and the Mathieu group M₁₀. Only S₆ and PGL(2,𝔽9) are semilinear groups; M₁₀ is a non-split extension.
• PΣL(2,𝔽16) = PΓL(2,𝔽16) — order 4 · 4080 = 16 320.
• PΣL(3,𝔽4) — order 2 · 20 160 = 40 320, while PΓL(3,𝔽4) is three times larger (gcd(3, 3) = 3).

Part of the wider family of matrix groups: the projective quotient of ΣL inside PΓL.

Wikipedia: Projective linear group