PSU(3,3) - Knowlpedia

$\operatorname{PSU}_{3}(3)$ is the following finite group, with $n=3$ and $q=3$ :

\operatorname{PSU}_{3}(3)=\operatorname{SU}_{3}(3)/\{\lambda I:\lambda^{3+1}=1,\ \lambda^{3}=1\}, \quad \operatorname{SU}_{3}(3)=\{A\in M_{3}(\mathbb F_{3^2}): (A^{(3)})^{\mathsf T}A=I,\ \det A=1\}.

Here $A^{(3)}$ is entrywise $3$ -th power. The matrix field has $3^2$ elements; its involution has fixed field of size $3$ . The scalar $\lambda$ belongs to $\mathbb F_{3^2}^\times$ ; multiplication is multiplication of scalar cosets.

Order and simplicity

Its order is

|\operatorname{PSU}_{3}(3)|=6048.

It is a nonabelian simple group. The exact order is 6,048.

Parameter convention

The parameter $q$ is the size of the fixed field, while matrices have entries in $\mathbb F_{q^2}$ . The Hermitian form is $h(v,w)=\sum_i v_i^q w_i$ . All nondegenerate Hermitian forms of this dimension over the finite field give isomorphic groups. Type ${}^2A_{n-1}$ has absolute rank $n-1$ ; it is not a unitary group over the real or complex numbers.

Small-group identification

This group is isomorphic to the index-two commutator subgroup $G_2(2)'$ .

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.
ATLAS of Finite Group Representations: U3(3) and G2(2) derived group, Heading and Standard generators: U3(3) = G2(2)′ has order 6048; U3(3):2 = G2(2).