Finite general linear group GL_n(q)

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

Order

The group order is $\prod_{j=0}^{n-1}(q^n-q^j)$ . For an invertible matrix, choose ordered linearly independent columns: after $j$ choices, precisely $q^j$ vectors are forbidden for the next column.

Simplicity conditions

In size one the group is $\mathbb F_q^\times$ , cyclic of order $q-1$ , and hence simple exactly when $q-1$ is prime. For size at least two and $q>2$ , the determinant has a proper nontrivial kernel, so simplicity fails. At $q=2$ , general and special linear groups coincide: size two gives $S_3$ , while every size at least three is simple.

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.