General linear groups

The general linear group GL(n,R) is the group of all invertible n × n matrices over a finite commutative ring R, under matrix multiplication. A matrix is invertible exactly when its determinant is a unit of R — over a field, when it is nonzero.

Over a finite field

GL(n,𝔽q) (q = pᵈ a prime power) is the automorphism group Aut(𝔽qn) of the n-dimensional vector space over 𝔽q.

• Order — count ordered bases of 𝔽qn: the i-th basis vector is any vector outside the span of the first i − 1, giving |GL(n,𝔽q)| = (qⁿ − 1)(qⁿ − q) ⋯ (qⁿ − qⁿ⁻¹) = qn(n−1)/2 ∏i=1n (qⁱ − 1).
• Center — the scalar matrices λI, λ ∈ 𝔽q×, isomorphic to Cq−1; quotienting by them gives PGL(n,𝔽q).
• Determinant — det: GL(n,𝔽q) → 𝔽q× is onto with kernel the special linear group SL(n,𝔽q), so GL/SL ≅ Cq−1.
• Sylow p-subgroup — the unitriangular group UT(n,𝔽q) (upper-triangular, 1s on the diagonal), of order qn(n−1)/2.

Small cases

• GL(1,𝔽q) ≅ Cq−1, the multiplicative group of the field.
• GL(2,𝔽2) ≅ S₃ (order 6), permuting the three nonzero vectors of 𝔽2².
• GL(3,𝔽2) ≅ PSL(2,𝔽₇), simple of order 168 — the second-smallest non-abelian simple group.
• GL(4,𝔽2) ≅ A₈, of order 20 160.
• Over 𝔽2 every determinant is 1, so GL(n,𝔽2) = SL(n,𝔽2) = PSL(n,𝔽2) — simple for every n ≥ 3.

Over the ring ℤ/m

GL(n,ℤ/m) is the group of n × n matrices over ℤ/m whose determinant is a unit of ℤ/m — an integer coprime to m. For prime m = p it is GL(n,𝔽p), so new groups start at m = 4.

• Prime-power modulus — reducing entries mod p maps GL(n,ℤ/pk) onto GL(n,𝔽p); the kernel, the matrices I + pM, has order p(k−1)n², so |GL(n,ℤ/pk)| = p(k−1)n² · |GL(n,𝔽p)|. E.g. |GL(2,ℤ/4)| = 2⁴ · 6 = 96 — not the same group as GL(2,𝔽₄) of order 180: the ring matters, not just its size.
• Direct products split — matrices over a product ring decompose componentwise, Mₙ(R₁ × R₂) ≅ Mₙ(R₁) × Mₙ(R₂) with a matrix invertible exactly when both components are, so GL(n,R₁ × R₂) ≅ GL(n,R₁) × GL(n,R₂) — e.g. GL(2,𝔽₂ × 𝔽₂) ≅ GL(2,𝔽2) × GL(2,𝔽2), of order 6 · 6 = 36. For coprime a, b the Chinese remainder theorem gives ℤ/ab ≅ ℤ/a × ℤ/b, so every GL(n,ℤ/m) is the direct product of its prime-power pieces — e.g. GL(2,ℤ/12) ≅ GL(2,ℤ/4) × GL(2,𝔽₃), of order 96 · 48 = 4608.
• Local rings are the fundamental case — every finite commutative ring is a direct product of local rings, uniquely up to reordering, so every GL(n,R) splits into general linear groups over local rings: the fields 𝔽q, the prime-power ℤ/pk, and other local rings like 𝔽₂[ε] (ε² = 0).

Part of the wider family of matrix groups: SL, PGL, PSL, and the classical, affine, and semilinear families.

Wikipedia: General linear group