Direct Systems and Direct Limit Groups Informal guided reading supervised by Dr Moty Katzman

Jai Adam Schrem

Based on Chapter 1 from Derek J.S. Robinson’s “A Course in the Theory of Groups”

23rd September 2020

Jai Adam Schrem Direct Systems and Direct Limit Groups Notation and Definitions

Let Λ be a set, with λ, µ, ν ∈ Λ. Definition: Equivalence Relation An equivalence relation is a binary relation between elements of a set Λ that is Reflexive: λ = λ Symmetric: If λ = µ, then µ = λ Transitive: If λ = µ and µ = ν, then λ = ν for any λ, µ, ν ∈ Λ.

Definition: Equivalence Classes We can partition the set Λ into subsets of elements that are equivalent to one another. These subsets are called equivalence classes. Write [λ] for the equivalence class of element λ ∈ Λ.

Jai Adam Schrem Direct Systems and Direct Limit Groups Notation and Definitions

Definition: Directed Set Equip set Λ with reflexive and transitive binary relation ‘≤’ that ‘compares’ elements pairwise. If the upper bound property

∀λ, µ ∈ Λ, ∃ν ∈ Λ such that λ ≤ ν and µ ≤ ν

is satisfied, we say the set Λ with binary relation ‘≤’ is directed.

Example: N directed by divisibility Equip N with the binary relation of divisibility, notated by ‘|’. Letting a, b, c ∈ N, we can easily see Reflexivity: Clearly a|a Transitivity: If a|b and b|c then a|c Upper bound property: For any a, b ∈ N we can compute the lowest common multiple, say m, so that a|m and b|m.

Jai Adam Schrem Direct Systems and Direct Limit Groups Direct System of Groups

Let Λ with binary relation ≤ be a directed set, with λ, µ, ν ∈ Λ. Let Gλ be a family of groups where λ ∈ Λ.  Let gλ ∈ Gλ ∈ Gλ : λ ∈ Λ . Let there be a family of group homomorphisms

µ αλ : Gλ → Gµ for λ ≤ µ where λ, µ ∈ Λ. µ Suppose that the homomorphisms αλ satisfy λ αλ is the identity map on Gλ µ ν ν αλαµ = αλ for any λ ≤ µ ≤ ν. Then we call the set

µ D = {Gλ, αλ : λ ≤ µ ∈ Λ} a direct system of groups.

Jai Adam Schrem Direct Systems and Direct Limit Groups Constructing the Direct Limit Group: Equivalence Classes

Technical note: we assume without loss of generality that the groups Gλ for λ ∈ Λ are disjoint, so that any g ∈ Gλ is not also contained in some other group Gµ. Where this is not true, we can replace any group in the family with an isomorphic copy.

Take the union of the elements in the family of groups

[  U = Gλ = g ∈ Gλ : λ ∈ Λ λ∈Λ

Now define an equivalence relation ∼ on the set U for gλ ∈ Gλ and gµ ∈ Gµ by gλ ∼ gµ whenever their images in Gν are equal:

ν ν gλ ∼ gµ ⇐⇒ αλ(gλ) = αµ(gµ) | {z } gν ∈Gν Let [g] be the equivalence class containing g, and let G be the set of all such equivalence classes

n  o G = g : g ∈ U

Jai Adam Schrem Direct Systems and Direct Limit Groups Constructing the Direct Limit Group: Group Operation

 Equip the set of equivalence classes G = [g]: g ∈ U with group operation

    ν ν  gλ gµ = αλ(gλ)αµ(gµ) Notice that:

Suppose gλ ∼ g˜λ˜ and gµ ∼ g˜µ˜. Then ∃ν ∈ Λ such that λ, λ,˜ µ, µ˜ ≤ ν. Thus αν (g ) = αν (g ) and αν (g ) = αν (g ) λ λ λ˜ λ˜ µ µ µ˜ µ˜ and the group operation is well-defined.   The identity element in the group is 1G = 1Gλ . This equivalence class contains all identity elements in U, because µ 1λ ∈ ker(αλ) for any λ ≤ µ ∈ Λ. Since inverses, closure and associativity hold in each group Gλ, they also hold in G.

Jai Adam Schrem Direct Systems and Direct Limit Groups The Pr¨uferGroup of Type p∞ Technical note: Here we take the directed set Λ to be N ordered in the usual way.

Let p be a fixed prime number, and let Gi = hxi i be a cyclic group with order pi . Define injective homomorphisms

i+1 σi : Gi → Gi+1 p xi 7→ xi+1 2 2p xi 7→ xi+1 . . pi pi+1 1i = xi 7→ xi+1 = 1i+1

k i+1 i+2 k with σi = σi ◦ σi+1 ◦ ... ◦ σk−1 for i < k. The limit of this direct system is an infinite abelian p-group called the Pr¨ufergroup of type p∞. It is a subgroup of the circle group.

Jai Adam Schrem Direct Systems and Direct Limit Groups Q under Addition as a Direct Limit Group

Let Λ be N ordered by divisibility, denoted ‘|’. 1 1 For q ∈ Λ, let Gq = h q i be the infinite cyclic group generated by q . Construct homomorphisms for q|p:

p βq : Gq → Gp p q˜ q˜ 7→ q q p For any q|r and p|r, define an equivalence relation ∼ on S  q˜ U = q∈Λ Gq = q ∈ Gq : q ∈ Λ by: q˜ p˜ q˜ p˜ ∼ ⇐⇒ βr = βr q p q q p p Equip the resulting equivalence classes with a group operation: r r r r hq˜i hp˜i  q˜ p˜  q˜ p˜  q˜ + p˜ + = βr + βr = q + p = q p q p q q p p r r r

Jai Adam Schrem Direct Systems and Direct Limit Groups