Free Groups — Mathematics Knowledge System

Overview

The free group F(S) on a set S is the 'most unconstrained' group generated by S: its elements are reduced words (strings of generators and their inverses with no adjacent cancellations), and no relations hold among the generators except those forced by the group axioms themselves. Every group generated by S is a quotient of F(S) — this is exactly what makes presentations ⟨S | R⟩ = F(S)/N(R) work, since any relations you want to impose are added on top of the free group as the baseline. Free groups are infinite whenever |S| ≥ 1 — even F({a}) with a single generator is isomorphic to the infinite cyclic group ℤ — and F(S) for |S| ≥ 2 is a rich source of non-abelian, torsion-free examples used throughout combinatorial and geometric group theory.

Intuition

Picture words built from letters a, b, a⁻¹, b⁻¹ where the only simplification allowed is cancelling a letter next to its own inverse (like turning 'abb⁻¹a' into 'aa'), and no other simplification is ever permitted — ab is never equal to ba unless you specifically want that. That total absence of imposed relations is exactly what 'free' means: F(S) is the baseline group generated by S with nothing extra assumed, and it's precisely this universal property (any assignment of the generators to elements of another group extends uniquely to a homomorphism) that lets you build any group at all as a quotient F(S)/N by choosing the right relations N to impose.

Formal Definition

Definition

For a set S, the free group F(S) consists of all reduced words in the alphabet S ∪ S⁻¹ (formal inverses), under concatenation followed by cancellation:

F(S) = \{\, \text{reduced words in } S \cup S^{-1} \,\}

Elements: reduced words

s s^{-1} = s^{-1}s = e \quad \text{(free cancellation, no other relations)}

Only relation: trivial cancellation

\text{Universal property: any map } S \to G \text{ extends uniquely to a homomorphism } F(S) \to G

Universal property of free groups

F(S_1) \cong F(S_2) \iff |S_1| = |S_2|

Free groups classified by rank (cardinality of generating set)

Properties

Universal property

\text{Any function } S \to G \text{ (for } G \text{ a group) extends uniquely to a group homomorphism } F(S) \to G.

Example: This is why every group is a quotient of some free group — map generators of G to themselves and extend.

Torsion-free

\text{Every nontrivial element of } F(S) \text{ has infinite order (no element other than } e \text{ satisfies } w^n = e \text{ for finite } n\text{).}

Rank

\text{The cardinality of a free generating set } S \text{ is an invariant of } F(S)\text{, called its rank.}

Example: F({a}) has rank 1 and is isomorphic to (\mathbb{Z}, +); F({a,b}) has rank 2 and is non-abelian.

Nielsen–Schreier theorem

\text{Every subgroup of a free group is itself free (though possibly of larger, even infinite, rank).}

Worked Examples

Adjacent b, b⁻¹ cancel directly.
$a\, b\, b^{-1} a^{-1} b = a\, a^{-1} b$
Now adjacent a, a⁻¹ cancel.
$a\, a^{-1} b = b$

Answer: The reduced word is simply b.

Practice Problems

Difficulty 4/10

Reduce the word a⁻¹ a b a a⁻¹ in F({a,b}) to its simplest form.

Difficulty 5/10

Every group G with generating set S is a quotient of F(S). Use this fact plus the fact that ℤ₆ is generated by one element to identify ℤ₆ as a quotient of a specific free group.

Difficulty 6/10

Using the universal property of F({a,b}), explain why there exists a (unique) homomorphism φ: F({a,b}) → D₄ sending a ↦ r (a 90° rotation) and b ↦ s (a reflection), without needing to check any relations first.

Common Mistakes

Common Mistake

Believing free groups are 'simple' because they have a simple description (just reduced words).

Free groups of rank ≥ 2 have extremely rich subgroup structure — by the Nielsen–Schreier theorem every subgroup is free, but subgroups of a rank-2 free group can themselves have infinite rank.

Quiz

Summary

F(S) consists of all reduced words in S ∪ S⁻¹, with only trivial cancellations (ss⁻¹ = e) allowed — no other relations.
Universal property: any map from S into a group G extends uniquely to a homomorphism F(S) → G, which is why every group is a quotient of a free group.
F(S) is torsion-free, classified up to isomorphism by rank |S|, and (Nielsen–Schreier) every subgroup of a free group is itself free.

References

WebsiteWikipedia — Free group