PSL(2,q) - Knowlpedia

For prime powers $q$ and fixed matrix size $2$ , $\operatorname{PSL}_{2}(q)$ is the finite group defined by

\operatorname{PSL}_{2}(q)=\operatorname{SL}_{2}(\mathbb F_{q})/\{\lambda I:\lambda\in\mathbb F_{q}^\times,\ \lambda^{2}=1\}, \qquad \operatorname{SL}_{2}(\mathbb F_{q})=\{A\in M_{2}(\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}_{2}(q)|=\frac{q(q^{2}-1)}{\gcd(2,q-1)}.

It is simple exactly when q ≥ 4; q = 2 and q = 3 are the two nonsimple cases.

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.