Free group
A free group is a group in which every element can be written uniquely as a reduced word in some set of generators, with no relations imposed between them beyond the group axioms themselves. It is the most unconstrained group possible on a given generating set — the group that results when one takes a set of symbols and declares them generators without adding any equations. The free group on a set \(S\), denoted \(F(S)\), is the universal group generated by \(S\): every function from \(S\) to a group \(G\) extends uniquely to a homomorphism \(F(S) \to G\). This universal property makes free groups the atomic building blocks of group theory, just as vector spaces are the atomic building blocks of linear algebra.
Structure and Construction
The elements of a free group are reduced words: finite strings of generators and their inverses, with no cancellation possible. For example, in the free group on two generators \(a\) and \(b\), the word \(aba^{-1}b^2\) is reduced, but \(abb^{-1}a\) reduces to \(a^2\). The group operation is concatenation followed by reduction. The empty word is the identity. This combinatorial description is precise, but it conceals the group's geometry: the Cayley graph of a free group is a tree — an infinite, branching, contractible graph with no cycles. This tree structure is not incidental; it is the geometric signature of freedom. A group is free if and only if it acts freely on a tree, a characterization that bridges algebra and geometry through Bass-Serre theory.
The rank of a free group is the cardinality of any free generating set. Unlike vector spaces, where every basis has the same cardinality by dimension theory, the fact that every free generating set of a group has the same size is a theorem (the Nielsen-Schreier theorem for finite rank, generalized by the Grushko theorem). The rank is the fundamental invariant of a free group, and the classification of finitely generated free groups up to isomorphism is trivial: two are isomorphic if and only if they have the same rank.
Subgroups and the Nielsen-Schreier Theorem
One of the most remarkable theorems in combinatorial group theory is the Nielsen-Schreier theorem: every subgroup of a free group is itself free. This is not obvious. The analogous statement for abelian groups is false (subgroups of free abelian groups are free abelian, but quotients need not be), and for general groups it is catastrophically false. The theorem was first proved by Jakob Nielsen for finitely generated subgroups and extended by Otto Schreier to the general case. The proof proceeds by analyzing the action of the subgroup on the Cayley tree: the quotient graph is a covering space, and covering spaces of trees are trees, hence the subgroup acts freely on a tree and is therefore free.
The theorem has a quantitative refinement. If \(F\) is a free group of rank \(n\) and \(H\) is a subgroup of finite index \(d\), then the rank of \(H\) is \(d(n-1) + 1\). This formula reveals that free groups are, in a precise sense, negatively curved: they have more subgroups than abelian groups of the same rank, and their subgroup growth is exponential. This growth is the algebraic shadow of the tree's exponential branching.
Free Groups and Presentations
Every group can be presented as a quotient of a free group. A group presentation \(\langle S \mid R \rangle\) consists of a generating set \(S\) and a set of relators \(R\) — words in \(F(S)\) that are set equal to the identity. The group defined by the presentation is \(F(S) / N(R)\), where \(N(R)\) is the normal closure of \(R\). This construction makes free groups the raw material from which all groups are carved. The Word problem for groups — the algorithmic question of whether a given word represents the identity — is trivial for free groups (reduce and check if empty), but undecidable in general. The boundary between free and non-free is, in this sense, the boundary between computability and uncomputability in group theory.
Free groups also sit at the foundation of Algebraic topology. The fundamental group of a bouquet of circles — a graph with one vertex and \(n\) edges — is the free group on \(n\) generators. This is the content of the Seifert-van Kampen theorem applied to a graph. Every graph of groups, in the sense of Bass-Serre theory, is built from free groups and amalgamated products, making free groups the base case of a vast structural theory.
The free group is not merely an algebraic curiosity. It is the primordial group — the group before relations, before constraints, before structure. Every equation we impose on a group is a wound inflicted on a free group; every classification theorem is an autopsy. To understand freedom in group theory is to understand what constraints cost. And the cost, measured in computability, geometry, and growth, is the entire content of the subject.