TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Free groups are torsion-free

Statement

Every free group is torsion-free: if $g$ is not the identity and $n \geq 1$ is a natural number, then $g^{n}$ is not the identity. Equivalently, every nonidentity element has infinite order in the sense of The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity.

Facts & Assumptions

Given: A free group $F$ on a set $X$ , a nonidentity element $g \in F$ , and a natural number $n \geq 1$ .

[L1]

Every nonempty reduced word has the form $t c t^{- 1}$ with $c$ nonempty and cyclically reduced (Every nonempty reduced word has the form $t c t^{- 1}$ with $c$ nonempty and cyclically reduced).

[L2]

The reduced words on $X ⊔ X^{- 1}$ form a group when the product of reduced words is their concatenation followed by free reduction, and the map sending $x \in X$ to the one-letter word $x$ has the universal property of the free group on $X$ (Reduced words form the free group on an alphabet).

[L3]

Free groups on the same set are uniquely isomorphic compatibly with their generators (Free groups on the same set are uniquely isomorphic compatibly with their generators).

[F1]

A reduced word is cyclically reduced when it is empty or its first letter is not the formal inverse of its last letter (Cyclically reduced words).

[F2]

An element $g$ has infinite order exactly when no positive natural power of $g$ is the identity (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity).

Proof

technique · direct

1.1

First work in the reduced-word model and let $w$ be a nonidentity element. Then $w$ is itself a nonempty reduced word, and [L1] gives a literal reduced factorisation $w = t c t^{- 1}$ with $c$ nonempty and cyclically reduced.

L1L2given

2.1

For every $n \geq 1$ , the literal concatenation $c^{n}$ is reduced and nonempty: each copy is reduced, and the seam between consecutive copies does not cancel because the last letter of $c$ is not the inverse of its first.

F1step 1.1

3.1

In the product $w^{n}$ , the adjacent factors $t^{- 1} t$ cancel between copies, leaving $t c^{n} t^{- 1}$ ; its two outer seams are the same seams as in the reduced word $t c t^{- 1}$ , so it is reduced and nonempty by step 2.1, and [L2] therefore shows that $w^{n}$ is not the identity.

L2step 1.1step 2.1

4.1

The reduced-word model is a free group on $X$ by [L2], so [L3] gives a generator-compatible isomorphism from an arbitrary free group on $X$ onto it; transporting $g$ along that isomorphism, which preserves the identity and natural powers, step 3.1 gives $g^{n} \neq e$ .

L2L3step 3.1

5.1

Thus no nonidentity element has a positive power equal to the identity, so by [F2] every nonidentity element has infinite order and every free group, including the trivial free group on the empty set, is torsion-free.

F2step 4.1∎

Depends on

Used by

A strict C prime(1/6) presentation that is a free group of rank six Example
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Dependency tree · two levels

28 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

Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial Group Theory (standard reference, not scraped)