PSL(1,q) - Knowlpedia

For prime powers $q$ and fixed matrix size $1$ , $\operatorname{PSL}_{1}(q)$ is the finite group defined by

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

This extends the projective special linear convention to matrix size one; multiplication is multiplication of scalar cosets. Here $\mathbb F_{q}$ is the field with $q$ elements.

Order and simplicity

Its order is

|\operatorname{PSL}_{1}(q)|=1.

It is the trivial group, so it is not simple under the nontrivial-group convention. The exact order is 1.

Parameter convention

The determinant-one condition forces the sole matrix entry to be 1. The resulting group has one element. This is a degenerate matrix-size case, not an additional irreducible Lie-type family.

References

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