General Linear Group - Over Finite Fields

Over Finite Fields

If F is a finite field with q elements, then we sometimes write GL(n,q) instead of GL(n,F). When p is prime, GL(n,p) is the outer automorphism group of the group Zn
p, and also the automorphism group, because Zn
p is Abelian, so the inner automorphism group is trivial.

The order of GL(n, q) is:

(qn − 1)(qn − q)(qn − q2) … (qn − qn−1)

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the kth column can be any vector not in the linear span of the first k − 1 columns. In q-analog notation, this is

For example, GL(3,2) has order (8 − 1)(8 − 2)(8 − 4) = 168. It is the automorphism group of the Fano plane and of the group Z3
2, and is also known as PSL(2,7).

More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to the Weil conjectures.

Note that in the limit q ↦ 1 the order of GL(n,q) goes to 0! — but under the correct procedure (dividing by (q-1)^n) we see that it is the order of the symmetric group (See Lorscheid's article) — in the philosophy of the field with one element, one thus interprets the symmetric group as the general linear group over the field with one element: S_n ≅ GL(n,1).