Stable general linear group over a field

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Definition

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

  • Define K^\omega as the direct sum of countably many copies of K. GL(K) is defined as the group of those linear automorphisms of K^\omega that fix pointwise all but finitely many of the copies.
  • Consider the general linear groups GL_n(K), with a natural inclusion map GL_n(K) \to GL_{n+1}(K) that sends A to the matrix with block description \begin{pmatrix}A & 0 \\ 0 & 1 \\\end{pmatrix}. GL(K) 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.