PSL(2,2) - Knowlpedia

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

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

This is the projective special linear group; multiplication is multiplication of scalar cosets. Here $\mathbb F_{2}$ is the field with $2$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{2}(2)|=6.

It is not simple: PSL(2,2) is isomorphic to S₃. The exact order is 6.

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 faithful action on the three points of the projective line gives $\operatorname{PSL}_2(2)\cong S_3$ .

References

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