General linear group over a field

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

This article is about the unit group (group of invertible elements) in the following ring/monoid: matrix ring

Definition

In terms of dimension

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

$G L (n, k)$ 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.
$G L (n, k)$ is the group of all invertible $n \times n$ matrices with entries over $k$

In terms of vector spaces

Let $V$ be a $k$ -vector space (which may be finite or infinite-dimensional). The general linear group over $V$ , denoted $G L (V)$ , is the group of all vector space automorphisms from $V$ to itself.

Note that when $V = k^{n}$ , this reduces to the definition $G l (n, k)$ . Further, since $G L (V) ≅ G L (W)$ for $V ≅ W$ , and since any two vectro spaces of the same dimension are isomorphic, the $G L (n, k)$ s cover all general linear groups corresponding to finite-dimensional vector spaces.

As a map

As a functor from fields to groups

If we fix $n$ , we can think of $G L_{n}$ as a functor from the category of fields to the category of groups.

As an IAPS

Further information: GL IAPS

For a fixed field $k$ , the general linear groups $G L (n, k)$ form an IAPS of groups parametrized by $n$ . In other words, we naturally have concatenation maps:

$Φ_{m, n} : G L (m, k) \times G L (n, k) \to G L (m + n, k)$

This map takes a matrix $A$ of order $m$ and a matrix $B$ of order $n$ and putputs the block diagonal matrix with blocks $A$ and $B$ .

As a functor from fields to IAPSes

If we fix neither $n$ nor $k$ , we can view $G L$ as a functor from fields to the category of IAPSes of groups.

Relation with other properties

Weaker properties