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The torsion-free reflection of the integers direct sum a finite cyclic group

Example

For $n \geq 1$ , let $G_{n} = Z \times (Z / n Z)$ , the direct sum of its two displayed abelian factors. Its torsion subgroup is ${0} \times (Z / n Z)$ , so its torsion-free reflection is $G_{n} / Tor (G_{n}) ≅ Z .$ For $n = 1$ the finite cyclic factor and the torsion subgroup are both trivial, and the same formula holds.

Facts & Assumptions

Given: A natural number $n \geq 1$ and the two displayed groups.

[L1]

The external direct product has underlying set of pairs and componentwise operation (The external direct product $G \times H$ with componentwise multiplication).

[L2]

This componentwise operation makes the product a group and its coordinate projections are homomorphisms ( $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections).

[L3]

The full subcategory of torsion-free abelian groups is reflective in $A b$ , with reflector $G \mapsto G / Tor (G)$ and unit the quotient map (Torsion-free abelian groups form a reflective full subcategory of abelian groups).

[L4]

For a full subcategory, a reflector with its adjunction is equivalently a specified universal arrow $(R_{C}, η_{C})$ from each object $C$ to the inclusion, the specified arrows being the components of the reflection unit (A full subcategory is reflectively structured exactly when universal arrows are supplied at every ambient object).

Verification

technique · direct

1.1L1L2algebra

By [L1] and [L2], $G_{n}$ has componentwise addition. Every $(0, \overset{ˉ}{a})$ is killed by $n$ . Conversely, if a nonzero integer $k$ kills $(z, \overset{ˉ}{a})$ , then $k z = 0$ in $Z$ , so $z = 0$ . Hence $Tor (G_{n}) = {0} \times (Z / n Z)$ .

2.1step 1.1L2algebra

The first projection $π_{1} : G_{n} \to Z$ is a homomorphism by [L2], and it is surjective because $π_{1} (z, 0) = z$ for every $z \in Z$ . Its kernel is ${(0, \overset{ˉ}{a})} = Tor (G_{n})$ by step 1.1, so the induced map $G_{n} / Tor (G_{n}) \to Z$ , $(z, \overset{ˉ}{a}) + Tor (G_{n}) \mapsto z$ , is a well-defined surjective homomorphism, and it is injective because $z = 0$ forces $(z, \overset{ˉ}{a}) \in Tor (G_{n})$ . It is therefore an isomorphism $G_{n} / Tor (G_{n}) ≅ Z$ .

3.1step 1.1step 2.1L3L4

By [L3] the reflector sends $G_{n}$ to $G_{n} / Tor (G_{n})$ with unit the quotient map, and by the equivalence in [L4] that unit is a universal arrow from $G_{n}$ to the inclusion: every homomorphism $f : G_{n} \to H$ with $H$ torsion-free factors uniquely through it. Step 2.1 identifies the target of that quotient map with $Z$ . Thus the computed object has the reflection's universal property.

4.1step 1.1step 2.1step 3.1∎

When $n = 1$ , $Z / n Z$ is the trivial group, so step 1.1 gives the zero torsion subgroup and steps 2.1–3.1 remain valid without dividing by a nontrivial integer.

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