Affine special linear groups

The affine special linear group ASL(n,𝔽q) is the group of affine maps x ↦ Ax + b of 𝔽qn whose linear part has determinant 1 — the translations extended by SL:

ASL(n,𝔽q) = 𝔽qn ⋊ SL(n,𝔽q) (a semidirect product).

• Order — |ASL(n,𝔽q)| = qⁿ · |SL(n,𝔽q)| = qⁿ · |GL(n,𝔽q)| / (q − 1).
• Inside AGL — x ↦ Ax + b sending A ↦ det A is a homomorphism AGL(n,𝔽q) → 𝔽q×, onto, with kernel ASL(n,𝔽q): so AGL / ASL ≅ Cq−1, an index-q−1 subgroup.
• Characteristic 2 — every determinant over 𝔽2 is 1, so ASL(n,𝔽2) = AGL(n,𝔽2).
• Action — for n ≥ 2 it is already 2-transitive on the qⁿ points: SL(n,𝔽q) is transitive on the non-zero vectors. For n = 1 the linear part dies (SL(1,𝔽q) = 1) and ASL(1,𝔽q) = (𝔽q, +) is just the translations, elementary abelian of order q.
• Volume — det A = 1 says exactly that the map preserves the standard alternating n-form of 𝔽qn, the finite-field analogue of a volume-preserving affine map.

Small cases

• ASL(2,𝔽2) = AGL(2,𝔽2) ≅ S₄ (order 24).
• ASL(2,𝔽3) — order 9 · 24 = 216, index 2 in AGL(2,𝔽3): the Hessian group of the affine plane AG(2,3), solvable, with point stabilizer SL(2,𝔽3) = Q₈ ⋊ C₃.
• ASL(2,𝔽4) — order 16 · 60 = 960; the point stabilizer SL(2,𝔽4) ≅ A₅ makes it non-solvable.
• ASL(3,𝔽2) = AGL(3,𝔽2) — order 8 · 168 = 1344.

Part of the wider family of matrix groups: the determinant-1 part of AGL, and AΣL is the same construction allowing a field twist.

Wikipedia: Affine group