TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-11

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The abelianisation of a free group on $X$ is a free abelian group on $X$

Statement

Let $(F (X), i)$ be a free group on $X$ , let $q : F (X) \to F (X)^{a b}$ be the abelianisation map, and put $i_{a b} = q \circ i$ . Then $(F (X)^{a b}, i_{a b})$ is a free abelian group on $X$ .

Facts & Assumptions

Given: A free group $(F (X), i)$ , its quotient $F (X)^{a b} = F (X) / [F (X), F (X)]$ , its canonical quotient map $q$ , and $i_{a b} = q \circ i$ .

[L1]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $[G, G] \subseteq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

[L2]

If a homomorphism $f : G \to H$ kills a normal subgroup $N$ , then it factors uniquely through $G / N$ (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

[F1]

A free abelian group on $X$ is an abelian group $A (X)$ with a map from $X$ such that every function from $X$ to an abelian group extends uniquely to a homomorphism from $A (X)$ (Free abelian group on a set).

[F2]

The commutator subgroup $[G, G]$ is the subgroup generated by all commutators $[g, h]$ (Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ ).

Proof

technique · constructive

1.1

Taking $N = [F (X), F (X)]$ in [L1] shows that $F (X)^{a b}$ is abelian.

L1given

1.2

Let $A$ be an abelian group and $u : X \to A$ a function; the free-group property gives a unique homomorphism $f : F (X) \to A$ extending $u$ , and $f ([g, h]) = f (g) f (h) f (g)^{- 1} f (h)^{- 1} = e_{A}$ because $A$ is abelian, so every commutator lies in the subgroup $\ker f$ ; since $[F (X), F (X)]$ is generated by those commutators by [F2], minimality gives $[F (X), F (X)] \subseteq \ker f$ .

F2given

2.1

By [L2], construct a homomorphism $\overline{f} : F (X)^{a b} \to A$ with $f = \overline{f} \circ q$ ; then $\overline{f} \circ i_{a b} = \overline{f} \circ q \circ i = f \circ i = u$ .

L2step 1.2construct

3.1

If $h : F (X)^{a b} \to A$ also extends $u$ , then $h \circ q$ and $\overline{f} \circ q$ are homomorphisms $F (X) \to A$ agreeing with $u$ on $X$ , so free-group uniqueness makes them equal; both $h$ and $\overline{f}$ therefore factor the same map through the quotient, and uniqueness in [L2] gives $h = \overline{f}$ .

L2step 2.1given

4.1

Steps 1.1, 2.1, and 3.1 give the abelian target, extension, and uniqueness clauses in [F1], so $F (X)^{a b}$ is free abelian on $X$ ; for $X = \emptyset$ both universal properties yield the trivial group.

F1step 1.1step 2.1step 3.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Free abelian group on a set.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard Elman, Lectures on Abstract Algebra, §18 (standard reference, not scraped)
John McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)

The abelianisation of a free group on XX is a free abelian group on XX