# Stable general linear group over a field

From Groupprops

## Definition

Let be a field. The **stable general linear group** over , denoted , is defined in the following equivalent ways:

- Define as the direct sum of countably many copies of . is defined as the group of those linear automorphisms of that fix pointwise all but finitely many of the copies.
- Consider the general linear groups , with a natural inclusion map that sends to the matrix with block description . is defined as the direct limit of this sequence of groups with homomorphisms.

Note that the stable general linear group is a *proper* subgroup of the general linear group of countable degree over a field.