PSU(2,q) - Knowlpedia

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

\operatorname{PSU}_{2}(q)=\operatorname{SU}_{2}(q)/\{\lambda I:\lambda^{q+1}=1,\ \lambda^{2}=1\}, \quad \operatorname{SU}_{2}(q)=\{A\in M_{2}(\mathbb F_{q^2}): (A^{(q)})^{\mathsf T}A=I,\ \det A=1\}.

Here $A^{(q)}$ is entrywise $q$ -th power. The matrix field has $q^2$ elements; its involution has fixed field of size $q$ . The scalar $\lambda$ belongs to $\mathbb F_{q^2}^\times$ ; multiplication is multiplication of scalar cosets.

Order and simplicity

Its order is

|\operatorname{PSU}_{2}(q)|=\frac{q(q^{2}-1)}{\gcd(2,q+1)}.

It is simple exactly when q ≥ 4, since PSU(2,q) is isomorphic to PSL(2,q).

Parameter convention

The matrices have entries in $\mathbb F_{q^2}$ , with involution $x\mapsto x^q$ , but this rank-one group is isomorphic to $\operatorname{PSL}_2(q)$ . The $A_1$ diagram has no nontrivial symmetry, so this does not define an additional twisted ${}^2A_1$ family.

References

Peter J. Cameron, Notes on Classical Groups, §§5.2–5.3, pp. 59–63; PSU(2,q), PSU(3,2), and the PSU(4,2) identification.
Robert A. Wilson, Classical groups (lecture notes), §3.1.7, pp. 5–6; §§3.2–3.4, pp. 7–17.