The free group on a set of generators is the largest possible group built from those generators — the one with no relations among them beyond the bare minimum required by the group axioms. Every other group built from the same generators is a quotient of the free group, obtained by imposing additional relations. Free groups are fundamental in combinatorial group theory, algebraic topology, and, as shown in the Banach-Tarski Paradox entry, even in set theory.
In an ordinary group, elements can satisfy unexpected relations — in the integers under addition, 1+1+1+...+1 (twelve times) equals 0 if you're working modulo 12. In the symmetric group on 3 elements, certain combinations of permutations return to the identity after a small number of steps. A free group has none of these unexpected coincidences — the only way two sequences of generators (and their inverses) are equal is if one can be reduced to the other by cancelling adjacent inverse pairs, like cancelling aa⁻¹ or b⁻¹b. Beyond that, nothing.
Free groups serve as the universal building blocks for all groups: every group is a quotient of some free group. This means that to understand all groups, it suffices in principle to understand free groups and the relations you can impose on them. This "generators and relations" perspective — called a group presentation — is the standard way to describe groups in combinatorial group theory and in algebraic topology.
Given a set S of generators, the free group F(S) consists of all finite sequences ("words") of elements of S and their formal inverses s⁻¹, modulo the equivalence relation generated by cancelling adjacent inverse pairs (replacing ...ss⁻¹... or ...s⁻¹s... by the empty sequence). These equivalence classes of "reduced words" form a group under concatenation followed by reduction. The empty word is the identity element.
The free group F(S) satisfies a precise universal property: for any group G and any function f: S → G, there is a unique group homomorphism F(S) → G extending f. This is the rigorous content of "the free group imposes no relations beyond the group axioms" — any assignment of the generators to group elements extends uniquely to a homomorphism, because there are no relations that could obstruct it.
A group presentation ⟨S | R⟩ describes a group as the quotient of the free group F(S) by the normal subgroup generated by a set of relators R — words in F(S) that are declared equal to the identity in the presented group. For example, ⟨a | a^n = e⟩ presents the cyclic group ℤ/nℤ; ⟨a, b | aba⁻¹b⁻¹ = e⟩ presents ℤ²; and ⟨a, b | ⟩ (no relations) presents F₂, the free group on two generators.
Given a group presentation ⟨S | R⟩ and two words in S, do they represent the same group element? This is the word problem for groups. For free groups, it is easily decidable (just reduce both words and check equality). But for groups given by arbitrary finite presentations, the word problem is undecidable in general — proved by Novikov (1955) and Boone (1957) independently, using techniques from computability theory. This was a landmark result: an algebraic problem shown to be undecidable.
Every subgroup of a free group is itself free — the Nielsen-Schreier theorem (Nielsen 1921, Schreier 1927). This elegant result is surprising: subgroups of groups need not inherit nice structural properties (subgroups of abelian groups are abelian, but subgroups of solvable groups need not be solvable in the same strong sense). That subgroups of free groups are free is a genuine structural rigidity result, with an elegant topological proof via covering spaces.
Bass-Serre theory (Serre, 1977) gives a deep geometric understanding of groups acting on trees. The fundamental theorem: a group acts on a tree (without inversions) if and only if it can be built from simpler groups via amalgamated products and HNN extensions — generalisations of free products. Free groups themselves arise as the fundamental groups of graphs, and Bass-Serre theory is the correct general framework for understanding how groups decompose along free-group-like structures.
As mentioned in the Banach-Tarski Paradox entry, the proof of that theorem exploits the fact that the rotation group of 3-dimensional space contains a free subgroup on two generators — F₂ sits inside SO(3). This free subgroup provides the "paradoxical decomposition" at the heart of the proof: the free group on two generators can be decomposed into two pieces, each of which is a copy of the whole free group (the "von Neumann paradox" for F₂). The fact that SO(3) contains F₂ — and that the plane rotation group SO(2) does not — is precisely why the Banach-Tarski paradox holds in 3 dimensions but not in 2.