PSL(2,5) - Knowlpedia

$\operatorname{PSL}_{2}(5)$ is the following finite group, with $n=2$ and $q=5$ :

\operatorname{PSL}_{2}(5)=\operatorname{SL}_{2}(\mathbb F_{5})/\{\lambda I:\lambda\in\mathbb F_{5}^\times,\ \lambda^{2}=1\}, \qquad \operatorname{SL}_{2}(\mathbb F_{5})=\{A\in M_{2}(\mathbb F_{5}):\det A=1\}.

This is the projective special linear group; multiplication is multiplication of scalar cosets. Here $\mathbb F_{5}$ is the field with $5$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{2}(5)|=60.

It is a nonabelian simple group. The exact order is 60.

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 $\operatorname{PSL}_2(3)\cong A_4$ .

Small-group identification

The exceptional isomorphism is $\operatorname{PSL}_2(5)\cong A_5\cong\operatorname{PSL}_2(4)$ . Its projective-line action has six points, so the five-letter identification is not that natural action.

References

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