PSL(n,q) - Knowlpedia

For integers $n\geq2$ and prime powers $q$ , excluding $(n,q)=(2,2),(2,3)$ , $\operatorname{PSL}_{n}(q)$ is the finite group defined by

\operatorname{PSL}_{n}(q)=\operatorname{SL}_{n}(\mathbb F_{q})/\{\lambda I:\lambda\in\mathbb F_{q}^\times,\ \lambda^{n}=1\}, \qquad \operatorname{SL}_{n}(\mathbb F_{q})=\{A\in M_{n}(\mathbb F_{q}):\det A=1\}.

This is the projective special linear group; multiplication is multiplication of scalar cosets. Here $\mathbb F_{q}$ is the field with $q$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{n}(q)|=\frac{q^{n(n-1)/2}\prod_{i=2}^{n}(q^i-1)}{\gcd(n,q-1)}.

It is simple for every admitted parameter: n ≥ 2, with (n,q) different from (2,2) and (2,3).

Parameter convention

Type $A_{n-1}$ uses matrix size $n$ , so the Lie rank is $n-1$ . The simple family omits $\operatorname{PSL}_2(2)\cong S_3$ and $\operatorname{PSL}_2(3)\cong A_4$ .

References

Peter J. Cameron, Notes on Classical Groups, §§2.4–2.5, pp. 19–21; Theorem 2.11.