General linear group over a finite field

This article defines a natural number-parametrized system of algebraic matrix groups. In other words, for every field and every natural number, we get a matrix group defined by a system of algebraic equations. The definition may also generalize to arbitrary commutative unital rings, though the default usage of the term is over fields.
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For the more general family of groups over *any* field, see General linear group over a field.

Definition

In terms of dimension (finite-dimensional case)

Let $n$ be a natural number and $k$ a finite field. The general linear group of degree $n$ over $k$, denoted $GL(n,k)$, is defined in the following equivalent ways:

• $GL(n,k)$ is the group of all invertible $k$-linear maps from the vector space $k^{n}$ to itself, under composition. In other words, it is the group of automorphisms of $k^{n}$ as a $k$-vector space.
• $GL(n,k)$ is the group of all invertible $n\times n$ matrices with entries over $k$

Arithmetic functions

$q=p^{a}$ here denotes the size of the finite field $k$.

Function	Value	Explanation
order	$(q^{n}-1)(q^{n}-q)...(q^{n}-q^{n-1})$	In order for the matrix to have non-zero determinant, the vector in the first column cannot be the zero vector, so we have $(q^{n}-1)$ choices for such a vector. The vector in the second column must be linearly independent with the first column, giving $(q^{n}-q)$ choices. In general, the vector in the ith column must be independent of the first, second, ..., i-1th column, giving $(q^{n}-q^{i-1})$ choices. Make such a choice of vector for each column.

Particular cases

Size of field	Order of matrices	Common name for the general linear group	Order of group	Comment
2	1	Trivial group	$1$	Trivial
3	1	Cyclic group:Z2	$2$	group of prime order
4	1	Cyclic group:Z3	$3$	group of prime order
5	1	Cyclic group:Z4	$4=2^{2}$	cyclic group
2	2	Symmetric group:S3	$6=2\cdot 3$	supersolvable but not nilpotent
3	2	General linear group:GL(2,3)	$48=2^{4}\cdot 3$	solvable but not supersolvable
4	2	Alternating group:A5	$60=2^{2}\cdot 3\cdot 5$	simple non-abelian group
5	2	General linear group:GL(2,5)	$480=2^{5}\cdot 3\cdot 5$	not solvable, has a simple non-abelian subquotient.
2	3	General linear group:GL(3,2)	$168=2^{3}\cdot 3\cdot 7$	simple non-abelian group