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 a priori 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 S_n should limit to an infinite symmetric group and the subgroups of a group G should limit to G, 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: the groups S_n don't limit to the permutation group of a countable set, S_\omega, 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

Let I be a set with a transitive, reflexive binary relation \preceq (a preorder). We call I a directed set if, for all i and j in I, there exists some k\in I such that i\preceq k and j\preceq k. Let \{G_i\} be a family of groups indexed by I with group homomorphisms f_{i,j}:G_i\to G_j for all i\preceq j in I such that

  1. f_{i,i}=\operatorname{id}_{G_i} for all i in I
  2. f_{j,k}\circ f_{i,j}=f_{i,k} for all i\preceq j\preceq k in I.

The pair \langle G_i,f_{i,j}\rangle is called a direct system, and we form the set. The direct limit of the direct system \langle G_i,f_{ij}\rangle is denoted by \varinjlim G_i and is defined on equivalence classes of the disjoint union of the G_i with x_i\sim x_j for x_i\in G_i and x_j\in G_j if f_{i,k}(x_i)=f_{j,k}(x_j), where k is the upper bound of i and j. That is,

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

For x_i\in G_i, x_j\in G_j, and upper bound k of i and j, we define the binary operation on \varinjlim G_i by setting [x_i]\cdot[x_j]=[f_{i,k}(x_i)f_{j,k}(x_j)], where the multiplication f_{i,k}(x_i)f_{j,k}(x_j) is performed in G_k. The operation is well defined by the compatibility condition on the f_{i,j}, and associativity follows from associativity in the G_i. Since each map f_{i,j} is a homomorphism, all identities lie in the same equivalence class, and this class forms the identity of \varinjlim G_i. Finally, the inverse of [x_i] for x_i\in G_i is simply [x_i^{-1}].

Like many categorical constructions, direct limits are unique in a strong sense: for two direct limits A and B of a direct system, there exists a unique isomorphism A\cong B.

Examples

  • The set of \mathbb{Z} under its usual order forms a directed set which indexes the family of finite symmetric groups S_n. With the usual embeddings f_{i,j}:S_i\to S_j, these groups and maps form a direct system with direct limit (isomorphic to) the subgroup of the symmetric group on countably many things S_\omega which contains permutations permutating only finitely many objects. If the integers \mathbb{Z} are viewed under the alternate partial order of divisibility, and only the approriate embeddings f_{i,j}:S_i\to S_j are kept, the same direct limit is formed.
  • For a prime number p, we form the direct system of factor groups \mathbb{Z}/p^n\mathbb{Z} with multiplication by phomomorphisms f_{i,i+1}:\mathbb{Z}/p^i\mathbb{Z} \rightarrow \mathbb{Z}/p^{i+1}\mathbb{Z}. Setting
f_{i,i+j}=f_{i+j-1,i+j}\circ\dots f_{i+1,i+2}\circ f_{i,i+1}

forms a direct system with direct limit called the Prüfer group \mathbb{Z}(p^\infty). One exists for each prime p and consists of all p^{\text{th}} roots of unity of all powers of p. This demonstrates that for totally ordered index sets, only the maps f_{i,i+1} need to be specified.

Properties

  • If \langle G_i,f_{i,j}\rangle is a direct system with index set I containing maximum element n, then \varinjlim G_i is (isomorphic to) G_n. Similarly, if there exists some i\in I such that for all j,k\geq i, G_j=G_k, then \varinjlim G_n is (isomorphic to) G_i.
  • The collection of finitely generated subgroups H_i of a given group G can be partially ordered by inclusion. Finite sets of finitely generated subgroups \{\langle X_1\rangle, \langle X_2\rangle,\dots \langle X_n\rangle\} are contained in the finitely generated subgroup \langle\cup X_i\rangle, so the index set is indeed directed. With the inclusion morphisms f_{i,j}:H_i\to H_j, the direct limit is simply (isomorphic to) G. 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.
  • Say\langle G_i,f_{i,j}\rangle is a direct system with index set (I,\preceq), and a \leq is an sub preorder on I such that (I,\leq) is a directed set and upper bounds under \leq are upper bounds under \preceq. Then the directed systems \langle G_i,f_{i,j}\rangle_{(I,\preceq)} and \langle G_i,f_{i,j}\rangle_{(I,\leq)} are isomorphic