Finite Chevalley central quotient

For a reduced irreducible root system $\Phi$ , prime $p$ , and $q=p^f$ with $f\geq1$ , the finite Chevalley central quotient is

G_\Phi(q)=\mathbf G_{\Phi,\mathrm{sc}}(\overline{\mathbb F}_p)^{F_q}/Z\bigl(\mathbf G_{\Phi,\mathrm{sc}}(\overline{\mathbb F}_p)^{F_q}\bigr).

Here $\mathbf G_{\Phi,\mathrm{sc}}$ is the split simply connected semisimple algebraic group with root system $\Phi$ , constructed from its Chevalley integral model. The Frobenius map $F_q$ raises coordinates to their $q$ -th powers; superscript $F_q$ means its fixed points. The denominator is the center of that finite group. Multiplication is multiplication of cosets, making this a quotient group.

Why the finite center matters

Taking fixed points of the adjoint algebraic group can produce a larger finite group than this quotient. For example, this convention gives $\operatorname{PSL}_n(q)$ , whereas adjoint type $A_{n-1}$ has fixed points $\operatorname{PGL}_n(q)$ . They need not coincide.

This is a construction, not an unconditional simplicity assertion. The small cases $A_1(2),A_1(3),B_2(2),G_2(2)$ require separate treatment.

References

Robert Steinberg, Lectures on Chevalley Groups (Yale, 1967), §3 construction; §4 Theorem 5, PDF pp. 52–54; §9 Theorems 24–25, PDF pp. 136–138.