Special linear groups

The special linear group SL(n,R) is the group of n × n matrices of determinant 1 over a finite commutative ring R — the kernel of the determinant det: GL(n,R) → R×, which is onto (scale one row), so SL(n,R) is normal in GL(n,R) with GL/SL ≅ R×.

Over a finite field

• Order — |SL(n,𝔽q)| = |GL(n,𝔽q)| / (q − 1) = qn(n−1)/2 ∏i=2n (qⁱ − 1).
• Center — the scalars λI with λⁿ = 1, cyclic of order gcd(n, q − 1); quotienting by it gives PSL(n,𝔽q).
• Generators — the transvections I + λE_ij (i ≠ j), the elementary row operations of determinant 1.
• Perfect — SL(n,𝔽_q) equals its own commutator subgroup for every n ≥ 2 and prime power q except SL(2,𝔽2) and SL(2,𝔽3).
• For odd q, −I is the unique involution of SL(2,𝔽q) — an involution has eigenvalues ±1 and determinant 1, forcing both to be −1.

Small cases

• SL(2,𝔽2) = GL(2,𝔽2) ≅ S₃ — over 𝔽2 every determinant is 1.
• SL(2,𝔽3): the binary tetrahedral group 2·A₄ of order 24 — the quaternion group Q₈ extended by C₃.
• SL(2,𝔽4) ≅ A₅ (order 60).
• SL(2,𝔽5): the binary icosahedral group 2·A₅ of order 120 — the smallest perfect group that is not simple.
• SL(3,𝔽2) = GL(3,𝔽2) ≅ PSL(2,𝔽₇), simple of order 168.

Over the ring ℤ/m

• The determinant-1 condition cuts the unit condition of GL(n,ℤ/m) down by the unit group: |SL(n,ℤ/m)| = |GL(n,ℤ/m)| / φ(m) (φ = number of units).
• Prime-power modulus — reducing mod p maps SL(n,ℤ/pk) onto SL(n,𝔽p) with kernel of order p(k−1)(n²−1), so |SL(n,ℤ/pk)| = p(k−1)(n²−1) · |SL(n,𝔽p)|; e.g. |SL(2,ℤ/4)| = 2³ · 6 = 48.
• Direct products split — as for GL, the determinant is 1 exactly when it is 1 in each component, so SL(n,ℤ/ab) ≅ SL(n,ℤ/a) × SL(n,ℤ/b) for coprime a, b, and the prime-power moduli are the fundamental case.
• Reduction from the integers — SL(n,ℤ) → SL(n,ℤ/m) is onto; for n = 2 its kernel is the principal congruence subgroup Γ(m) of the modular group SL(2,ℤ).

Part of the wider family of matrix groups; the determinant-1 subgroup of GL, with projective quotient PSL.

Wikipedia: Special linear group