Affine special semilinear groups

The affine special semilinear group AΣL(n,𝔽q) is the translations of 𝔽qn extended by ΣL(n,𝔽_q) — the maps x ↦ A·xσ + b with det A = 1:

AΣL(n,𝔽q) = 𝔽qn ⋊ ΣL(n,𝔽q), q = pd.

• Order — |AΣL(n,𝔽q)| = qⁿ · d · |SL(n,𝔽q)| = d · |ASL(n,𝔽q)|.
• Over a prime field — AΣL(n,𝔽p) = ASL(n,𝔽p).
• Inside AΓL — index q − 1, the determinant of the linear part being onto 𝔽q×; equal to AΓL(n,𝔽_q) when q = 2ᵈ — over 𝔽2 every determinant is 1.
• Action — for n ≥ 2 it is 2-transitive on the qⁿ points, SL(n,𝔽q) being transitive on the non-zero vectors.

The one-dimensional case

SL(1,𝔽q) = 1, so AΣL(1,𝔽q) is the translations extended by the Galois group alone — 𝔽q ⋊ Cd, of order qd. For q > 2 it is not 2-transitive: the stabilizer of 0 fixes the prime subfield pointwise, so it cannot move 1. It is a p-group exactly when d is a power of p.

• AΣL(1,𝔽4) ≅ D₄ (order 8): the Frobenius swaps the two primitive elements of 𝔽4, acting on (C2)² as a transposition.
• AΣL(1,𝔽8) ≅ C₂ × A₄ (order 24): as a module over the order-3 Frobenius, 𝔽8 splits into the fixed 𝔽2 and a 2-dimensional summand on which C3 acts irreducibly, and (C2)² ⋊ C3 ≅ A₄.
• AΣL(1,𝔽9) ≅ C3 × S3 (order 18): the Frobenius x ↦ x³ fixes 𝔽3 pointwise and inverts a complementary line.
• AΣL(1,𝔽16) — order 64, a 2-group.

Higher dimensions

• AΣL(2,𝔽4) — order 16 · 2 · 60 = 1920.
• AΣL(2,𝔽8) — order 64 · 3 · 504 = 96 768.
• AΣL(2,𝔽9) — order 81 · 2 · 720 = 116 640.

Part of the wider family of matrix groups: the determinant-1 part of AΓL, and ASL with a field twist.

Wikipedia: Semilinear map