In mathematics, a direct limit of groups is the direct limit of a direct system of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called finitary or stable groups, though this latter term normally means something quite different in model theory.

Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.

The notion of a direct limit captures many vague but intuitive ideas of "group limits": the finite Symmetric groups, in some sense. Under the direct limit construction, group families (symmetric groups, dihedral groups, general linear groups, etc) will generally limit to the finitary or stable subgroup of the corresponding infinite group, but do limit to its subgroup of permutations which permute only finitely many objects. We'll also often see that recovering a group as a direct limit of its subgroups can be done simply (and sometimes only) with its finitely generated subgroups. Direct limits have a more general definition in Category theory, which reduces to the definition below in the category of groups, and more generally, any concrete category.

Definition

(a preorder a directed set such that

1. .

is called a direct system, and we form the set. The direct limit and is defined on equivalence classes of the disjoint union. That is,

\varinjlim G_i \text{ has underlying set }\bigsqcup_{i\in I} G_i\bigg/\sim.

.

.

Examples

• under its usual order forms a directed set are kept, the same direct limit is formed.

• For a prime number, we form the direct system of factor groups forms a direct system with direct limit called the Prüfer group roots of unity need to be specified.

• Each family of classical groups forms a direct system, via inclusion of matrices. Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group. The Whitehead group of a ring (the first K-group. Stable homotopy groups of spheres are the stable groups associated with the suspension functor.

Properties

• .

• . An analogous result holds for rings, modules, algebras, etc. Note the requirement of finite generation may be weakened, as long as the index set remains directed. Often times, this is only possible if the entire group is included in the collection of subgroups.

• are isomorphic

See also

References