Finite general linear group GL_3(q)

Let $q=p^f$ , with $p$ prime and $f\geq1$ an integer, and choose the finite field $\mathbb F_q$ . The general linear group $\mathrm{GL}_{3}(q)$ is the group of invertible $3\times 3$ matrices over $\mathbb F_q$ , with matrix multiplication.

Order

The group order is $(q^3-1)(q^3-q)(q^3-q^2)$ . For an invertible matrix, choose ordered linearly independent columns: after $j$ choices, precisely $q^j$ vectors are forbidden for the next column.

Simplicity conditions

For $q>2$ , the determinant has a proper nontrivial kernel, the special linear group. For $q=2$ , every nonzero determinant equals one and the center is trivial, so $\mathrm{GL}_3(2)=\mathrm{SL}_3(2)=\mathrm{PSL}_3(2)$ , which is simple. Simplicity therefore occurs exactly at $q=2$ .

Field and category convention

The parameter $q$ must be a prime power. This is a finite abstract group; the field and its vector space supply a construction, rather than a vector-space structure on the matrix group itself.

References

Peter J. Cameron, Notes on Classical Groups, §2.1, Theorems 2.2–2.3, pp. 15–16: orders, determinant and scalar kernels; §2.4, Theorem 2.10, pp. 20–22: simplicity.