Knowlpedia

Definition

PSL(2,q²)

Determinant-one two by two matrices over F_(q²), modulo scalar matrices.

For a prime power qq, PSL⁡2(q2)\operatorname{PSL}_2(q^2) is the

PSL⁡2(q2)={A∈M2(Fq2):det⁡A=1}/{λI:λ∈Fq2×,λ2=1}.\operatorname{PSL}_2(q^2)=\{A\in M_2(\mathbb F_{q^2}):\det A=1\}/\{\lambda I:\lambda\in\mathbb F_{q^2}^\times,\lambda^2=1\}.

Multiplication is multiplication of scalar cosets. Here the matrix field has size q2q^2, not qq.

Order and simplicity

Its order is

∣PSL⁡2(q2)∣=q2(q4−1)gcd⁡(2,q2−1).|\operatorname{PSL}_2(q^2)|=\frac{q^2(q^4-1)}{\gcd(2,q^2-1)}.

It is simple for every admitted prime power q, since q² ≥ 4.

Orthogonal identification

The low-rank identification is P⁡Ω4−(q)≅PSL⁡2(q2)\operatorname{P}\Omega_4^-(q)\cong\operatorname{PSL}_2(q^2). For example q=2q=2 gives PSL⁡2(4)≅A5\operatorname{PSL}_2(4)\cong A_5.

References
  1. Peter J. Cameron, Notes on Classical Groups, §§2.4–2.5, pp. 19–21; Theorem 2.11.
  2. Peter J. Cameron, Notes on Classical Groups, §6, pp. 64–74, especially Theorems 6.3 and 6.6; §7 introduction, p. 75.