General semilinear groups

The general semilinear group ΓL(n,𝔽_q) is the group of all invertible semilinear maps of 𝔽_qⁿ: additive bijections f for which there is a field automorphism σ with f(λx) = λ^σ f(x). Every one is x ↦ A·x^σ for a unique A ∈ GL(n,𝔽_q) and σ, so

ΓL(n,𝔽_q) = GL(n,𝔽_q) ⋊ Gal(𝔽_q / 𝔽_p) ≅ GL(n,𝔽_q) ⋊ C_d, where q = p^d.

Order — |ΓL(n,𝔽_q)| = d · |GL(n,𝔽_q)|.
Over a prime field — Gal(𝔽_p / 𝔽_p) is trivial, so ΓL(n,𝔽_p) = GL(n,𝔽_p): new groups appear only for q = p^d with d ≥ 2.
The action — the Frobenius x ↦ x^p acts on GL(n,𝔽_q) entry by entry, A ↦ A^(p), and generates the complement C_d.
Inside a bigger linear group — a semilinear map is 𝔽_p-linear, so ΓL(n,𝔽_q) ≤ GL(nd,𝔽_p): it is the normalizer there of the group of 𝔽_q-scalars, whose centralizer is GL(n,𝔽_q) and whose normalizer quotient is the Galois group.

Dimension 1

ΓL(1,𝔽_q) = 𝔽_q^× ⋊ C_d — order d(q − 1), metacyclic, the normalizer of a Singer cycle in GL(d,𝔽_p).

ΓL(1,𝔽₄) ≅ S₃ — C₃ ⋊ C₂, order 6.
ΓL(1,𝔽₈) — C₇ ⋊ C₃, order 21.
ΓL(1,𝔽₉) ≅ SD₁₆, the semi-dihedral group of order 16: the Frobenius acts on 𝔽₉^× ≅ C₈ as x ↦ x³ — cubing, not inverting, which is exactly the semi-dihedral action.
ΓL(1,𝔽₁₆) — C₁₅ ⋊ C₄, order 60.

When is it a Frobenius group? φ^k fixes the subfield 𝔽_p^g pointwise, g = gcd(k, d), so it fixes no element of 𝔽_q^× but 1 exactly when p^g = 2. Hence ΓL(1,𝔽_q) is a Frobenius group precisely for q = 2ᵈ with d prime — ΓL(1,𝔽₄) and ΓL(1,𝔽₈) are, ΓL(1,𝔽₁₆) is not (its φ² fixes 𝔽₄).

Higher dimensions

ΓL(2,𝔽₄) — order 2 · 180 = 360.
ΓL(2,𝔽₉) — order 2 · 5760 = 11 520.

Quotienting by the scalars gives PΓL; the determinant-1 linear part gives ΣL, and adjoining translations gives AΓL.

Part of the wider family of matrix groups: the semilinear extension of GL.

Wikipedia: Semilinear map