Free Groups and Group Presentations

⚡ Key Takeaways

Free groups are groups with no relations among their generators beyond those forced by the group axioms; every group is a quotient of a free group. A group presentation $\langle X \mid R \rangle$ specifies a group by generators $X$ and relations $R$ , encoding it as a free group modulo the relations. They apply throughout combinatorial group theory, topology (fundamental groups), and algebra. The foundations were laid by Walther von Dyck (1882) and developed by Nielsen, Schreier, and others in the early 20th century.

💡 The Core Insight

Start with an alphabet $X$ . Allow yourself to write down every word you can, refusing to assume any coincidences. The only simplifications permitted are $x x^{-1} = e$ and $x^{-1} x = e$ . The resulting group is as "large" and "unconstrained" as possible on $X$ . To build any specific group, you then impose relations — equations declaring certain words equal to the identity. This two-step recipe (free group, then quotient by relations) describes every group whatsoever.

DefinitionWords, Reduction, and the Free Group

Let $X$ be a set (the alphabet or generating set). Form the set of symbols $X^{\pm} = X \cup X^{-1}$ , where $X^{-1} = \{ x^{-1} : x \in X\}$ is a disjoint copy. A word is a finite sequence

1

\quad w = x_{1}^{\epsilon_1} x_{2}^{\epsilon_2} \cdots x_{n}^{\epsilon_n}, \qquad x_i \in X,\ \epsilon_i \in \{+1, -1\},

Derivation hint: a word is just an element of the free monoid on $X^{\pm}$ ; $n \geq 0$ , with $n=0$ giving the empty word $e$ . A word is reduced if it contains no adjacent pair $x x^{-1}$ or $x^{-1} x$ . The free group $F(X)$ is the set of all reduced words, with the product of $u$ and $v$ defined by concatenating and then cancelling cancelling pairs until reduced:

2

\quad u \cdot v = \mathrm{reduce}(uv).

Derivation hint: well-definedness requires that reduction yields a unique result — guaranteed by the Diamond Lemma / van der Waerden's trick below. The rank of $F(X)$ is the cardinality $|X|$ . We write $F_n$ for the free group of rank $n$ , and $F_\infty$ for countably infinite rank.

DefinitionGroup Presentation

Given a set $X$ and a set $R$ of words in $X^{\pm}$ (the relators), the presentation $\langle X \mid R \rangle$ denotes the group

3

\quad \langle X \mid R \rangle \;=\; F(X) \,/\, N(R),

Derivation hint: $N(R)$ is the smallest normal subgroup of $F(X)$ containing $R$ — the <strong>normal closure</strong>, formed by taking all conjugates $w r^{\pm 1} w^{-1}$ of relators and all their products. The normal closure is explicitly

4

\quad N(R) = \Big\{\, \textstyle\prod_{i=1}^{k} w_i\, r_i^{\delta_i}\, w_i^{-1} \;:\; w_i \in F(X),\ r_i \in R,\ \delta_i = \pm 1,\ k \geq 0 \,\Big\}.

Derivation hint: this set is closed under product, inverse, and conjugation, and any normal subgroup containing $R$ must contain all such elements; hence it is the intersection of all normal subgroups containing $R$ . A group $G$ is finitely generated if it has a presentation with $|X| < \infty$ , and finitely presented if both $|X|, |R| < \infty$ . We often write relations as equations: a relator $r = ab a^{-1} b^{-1}$ corresponds to the relation $ab = ba$ .

TheoremUniversal Property of Free Groups

Let $\iota : X \hookrightarrow F(X)$ be the inclusion of generators. For every group $G$ and every function $f : X \to G$ , there exists a unique group homomorphism $\bar{f} : F(X) \to G$ with $\bar{f} \circ \iota = f$ :

5

\quad \bar{f}\big(x_1^{\epsilon_1} \cdots x_n^{\epsilon_n}\big) = f(x_1)^{\epsilon_1} \cdots f(x_n)^{\epsilon_n}.

Derivation hint: define $\bar f$ on words by applying $f$ symbol-by-symbol; reduction is respected because $f(x)f(x)^{-1}=e$ in $G$ ; uniqueness follows since $X$ generates $F(X)$ . This property characterizes $F(X)$ up to unique isomorphism, and is the defining feature of "freeness" in category theory: $F$ is left adjoint to the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$ .

TheoremUniversal Property of Presentations — von Dyck's Theorem

Let $G = \langle X \mid R \rangle$ with quotient map $\pi : F(X) \to G$ . If $H$ is any group and $f : X \to H$ a function such that $\bar f(r) = e_H$ for every $r \in R$ (i.e., the relations hold in $H$ ), then $f$ induces a unique homomorphism $\phi : G \to H$ with $\phi(\pi(x)) = f(x)$ :

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\quad \phi\big(\pi(w)\big) = \bar f(w), \qquad w \in F(X).

Derivation hint: $\bar f$ kills $R$ , hence kills $N(R) = \ker\pi$ , so $\bar f$ factors through $F(X)/N(R) = G$ by the first isomorphism theorem.

ProofReduction is Well-Defined (van der Waerden's Trick)

We must show that the group operation [2] is associative and that reduced words form a group. The subtle point is that a word may be reducible in several orders; we need the reduced form to be unique.

Step 1 — Define an action on reduced words. Let $W$ be the set of reduced words. For each symbol $a \in X^{\pm}$ , define a map $\sigma_a : W \to W$ by

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\quad \sigma_a(x_1 \cdots x_n) = \begin{cases} x_2 \cdots x_n & \text{if } x_1 = a^{-1}, \\ a\, x_1 \cdots x_n & \text{otherwise}.\end{cases}

Derivation hint: prepending $a$ either cancels a leading $a^{-1}$ or extends the word, always producing a reduced word.

Step 2 — Each $\sigma_a$ is a bijection with inverse $\sigma_{a^{-1}}$ : one checks $\sigma_{a^{-1}}\sigma_a = \mathrm{id}_W$ directly by cases. Hence each $\sigma_a$ lies in the symmetric group $\mathrm{Sym}(W)$ .

Step 3 — Map words to permutations. Extend to any word $w = a_1 \cdots a_n$ by $\Sigma_w = \sigma_{a_1} \circ \cdots \circ \sigma_{a_n}$ . Then $\Sigma_w$ depends only on the reduced form of $w$ , because inserting a cancelling pair $a a^{-1}$ inserts $\sigma_a \sigma_{a^{-1}} = \mathrm{id}$ , which changes nothing.

Step 4 — Recover the word. Applying $\Sigma_w$ to the empty word $e$ returns the reduced form of $w$ :

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\quad \Sigma_w(e) = \mathrm{reduce}(w).

Derivation hint: build up $w$ symbol by symbol from the right; each $\sigma_{a_i}$ prepends or cancels exactly as reduction would.

Step 5 — Conclude. The map $w \mapsto \Sigma_w$ sends reduced words injectively into $\mathrm{Sym}(W)$ (by Step 4) and is a homomorphism for concatenation. Hence reduced words inherit a well-defined, associative product, and form a group: the identity is $e$ and the inverse of $x_1 \cdots x_n$ is $x_n^{-1} \cdots x_1^{-1}$ . $\blacksquare$

∎

ProofEvery Group Has a Presentation

Let $G$ be any group. Choose $X = G$ as the generating set (or any generating set). The universal property [5] gives a surjective homomorphism $\pi : F(X) \twoheadrightarrow G$ (surjective because $X$ generates $G$ ). Let $R$ be a set of words generating $\ker \pi$ as a normal subgroup; e.g. take $R = \ker\pi$ itself. By the first isomorphism theorem,

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\quad G \;\cong\; F(X)/\ker\pi \;=\; \langle X \mid R \rangle.

Derivation hint: surjectivity gives $\mathrm{im}\,\pi = G$ ; quotienting $F(X)$ by the kernel recovers $G$ ; choosing $R$ to normally generate $\ker\pi$ realizes the kernel as $N(R)$ . Thus every group — finite or infinite — arises as a quotient of a free group. $\blacksquare$

∎

mathematician · Walther von Dyck | 1856–1934

A student of Felix Klein, von Dyck introduced group presentations in his 1882 paper "Gruppentheoretische Studien." He was the first to study groups abstractly via generators and defining relations, rather than as concrete permutation or matrix groups, and proved the universal mapping property now bearing his name (Theorem [6]). His work bridged Klein's geometric program with the emerging abstract theory; the "von Dyck groups" $\langle a,b \mid a^p, b^q, (ab)^r\rangle$ governing tessellations of the sphere, plane, and hyperbolic plane carry his name. He also edited Kepler's and Galileo's collected works and founded the Deutsches Museum in Munich.

🧠 Intuition

Picture the free group $F_2 = F(\{a,b\})$ as an infinite tree — specifically the Cayley graph, a 4-valent tree with no loops. Standing at the identity vertex, you may step in four directions: $a$ , $a^{-1}$ , $b$ , $b^{-1}$ . Each reduced word is a unique non-backtracking path from the root; "non-backtracking" is the reduced condition (don't immediately undo your last step). Because the graph is a tree, there is exactly one such path to each vertex — this is precisely why each group element equals exactly one reduced word. A relation corresponds to declaring that some loop in this tree should actually close up. Imposing $ab = ba$ glues paths together so that "go $a$ then $b$ " lands at the same place as "go $b$ then $a$ ," collapsing the tree into the grid $\mathbb{Z}^2$ . The more relations you add, the more the tree folds onto itself; the free group is the unfolded extreme where nothing is identified beyond the forced cancellations.

Figure

The Cayley graph of

F_2

is an infinite 4-valent tree; reduced words are non-backtracking paths from the identity.

```svg

```

ExampleComputing in

\langle a, b \mid a^2,\, b^3,\, (ab)^2 \rangle

Given: The presentation $G = \langle a, b \mid a^2,\, b^3,\, (ab)^2\rangle$ . We claim $G \cong S_3$ , the symmetric group on 3 letters (order 6).

Step 1 — Read the relations as equations.

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\quad a^2 = e,\qquad b^3 = e,\qquad (ab)^2 = e \;\Rightarrow\; abab = e.

Derivation hint: a relator $r$ being trivial means $r = e$ in $G$ .

Step 2 — Derive a commutation rule. From $abab = e$ and $a^2 = e$ (so $a^{-1}=a$ ):

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\quad abab = e \;\Rightarrow\; ba = (ab)^{-1} = b^{-1}a^{-1} = b^{-1}a = b^2 a.

Derivation hint: invert $ab$ using $(xy)^{-1}=y^{-1}x^{-1}$ ; then $a^{-1}=a$ , $b^{-1}=b^2$ .

Step 3 — Enumerate normal forms. Every element can be pushed to the form $b^i a^j$ with $i \in \{0,1,2\}$ , $j \in \{0,1\}$ , using $ab = b^2 a$ to move $a$ 's rightward:

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\quad G = \{\, e,\ b,\ b^2,\ a,\ ba,\ b^2 a \,\}, \qquad |G| \le 6.

Derivation hint: any word reduces to $b^i a^j$ ; the relations cap the count at $3 \times 2 = 6$ .

Step 4 — Realize in $S_3$ . Map $a \mapsto (1\,2)$ , $b \mapsto (1\,2\,3)$ . Check: $(1\,2)^2 = e$ , $(1\,2\,3)^3 = e$ , and $\big((1\,2)(1\,2\,3)\big)^2 = (2\,3)^2 = e$ . All relations hold, so von Dyck's theorem gives a surjection $G \twoheadrightarrow S_3$ , hence $|G| \geq 6$ .

Answer: $\boxed{G \cong S_3,\ |G| = 6}$ .

ExampleThe Free Abelianization

Given: Compute the abelianization of the free group $F_2 = \langle a, b\rangle$ , i.e. $F_2 / [F_2, F_2]$ .

Step 1 — Add commutator relations. The commutator subgroup is normally generated by $[a,b] = aba^{-1}b^{-1}$ . Adjoining this relator:

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\quad F_2^{\mathrm{ab}} = \langle a, b \mid aba^{-1}b^{-1}\rangle = \langle a, b \mid ab = ba\rangle.

Derivation hint: setting the commutator to $e$ forces $a,b$ to commute.

Step 2 — Recognize the structure. Once $a,b$ commute, every word equals $a^m b^n$ uniquely:

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\quad F_2^{\mathrm{ab}} \;\xrightarrow{\ \cong\ }\; \mathbb{Z}^2, \qquad a^m b^n \mapsto (m,n).

Derivation hint: commuting generators of infinite order give the free abelian group of rank 2; verify the map is a bijective homomorphism.

Step 3 — General rank. The same argument shows $F_n^{\mathrm{ab}} \cong \mathbb{Z}^n$ . Since $\mathbb{Z}^m \cong \mathbb{Z}^n$ iff $m=n$ , this proves rank is an invariant:

Answer: $\boxed{F_n^{\mathrm{ab}} \cong \mathbb{Z}^n,\ \text{so } F_m \cong F_n \iff m = n}$ .

⚠️ Relations are not unique, and "no relations" ≠ "no equalities"

A common error is to think a group has a canonical presentation. It does not: $\langle a \mid a^6\rangle$ , $\langle x,y \mid x^2, y^3, xy x^{-1} y^{-1}\rangle$ , and others can all present $\mathbb{Z}/6\mathbb{Z}$ . Different presentations of the same group are related by Tietze transformations (adding/removing redundant generators or relators). A second error: students say free groups have "no relations." More precisely, free groups have no relations beyond those required by the group axioms. The equation $a a^{-1} = e$ certainly holds in $F(\{a\})$ — that is forced. "Free" means no extra relations, not no equalities whatsoever.

CounterexampleSubgroups of free groups can have larger rank

One might guess that a subgroup of $F_n$ has rank $\le n$ . False. The commutator subgroup $[F_2, F_2]$ of $F_2$ is itself a free group of infinite rank. By the Nielsen–Schreier theorem, every subgroup of a free group is free, but the rank can balloon: a subgroup of index $k$ in $F_n$ is free of rank

15

\quad \mathrm{rank} = k(n-1) + 1,

Derivation hint: this is the Schreier index formula, derivable from Euler characteristic $\chi$ multiplicativity for covering spaces of a wedge of $n$ circles. So a finite-index subgroup of $F_2$ ( $n=2$ ) of index $k$ has rank $k+1$ , which grows without bound. Free groups are infinitely "self-similar."

connection

Free groups and presentations sit at a crossroads of algebra, topology, and logic.

Topology: By the Seifert–van Kampen Theorem, the fundamental group of a wedge of $n$ circles is exactly $F_n$ , and a CW-complex with one 0-cell, generators as 1-cells, and relators as 2-cells has fundamental group $\langle X \mid R\rangle$ . Presentations are 2-complexes.
Universal algebra: The free-forgetful adjunction (Theorem [5]) is the prototype of Free Objects in a Category; free monoids, free abelian groups, and free modules all follow the same pattern.
Quotient structure: Presentations are an instance of the First Isomorphism Theorem applied to $F(X) \to G$ , with relators normally generating the kernel.
Geometric group theory: Cayley Graphs turn group elements into vertices and generators into edges, making free groups into trees and giving a geometric notion of group structure.
Combinatorics: The Word Problem for Groups asks whether a given word equals the identity; for free groups it is trivially solvable by reduction, but for general presentations it can be undecidable.

🔬 Where this appears

⏳ History

- 1856 (Hamilton): The icosian calculus gives an early concrete presentation-like description of the symmetry group of the icosahedron via generators and relations.

- 1878 (Cayley): Arthur Cayley introduces the idea of representing a group by its multiplication structure and graphs (later Cayley graphs), and abstractly defines groups.

- 1882 (von Dyck): Walther von Dyck formalizes generators and defining relations, proving the universal property and effectively defining free groups as the relation-free case.

- 1908 (Tietze): Heinrich Tietze introduces Tietze transformations, showing any two finite presentations of the same group are connected by elementary moves, and poses the isomorphism problem.

- 1911 (Dehn): Max Dehn formulates the three fundamental decision problems — the word, conjugacy, and isomorphism problems — and solves the word problem for surface groups.

- 1921–1927 (Nielsen, Schreier): Jakob Nielsen and Otto Schreier prove that every subgroup of a free group is free, with Schreier supplying the index formula [15].

- 1936 (Todd, Coxeter): The Todd–Coxeter coset enumeration algorithm provides a systematic method to compute finite quotients from presentations.

- 1955 (Novikov), 1958 (Boone): Pyotr Novikov and (independently) William Boone prove the word problem is undecidable for general finitely presented groups — a landmark linking group theory and computability.

- 1987 (Gromov): Mikhael Gromov's theory of hyperbolic groups recasts free groups as the simplest hyperbolic groups, launching modern geometric group theory.

open-problem

The transition from a presentation to the group it presents hides deep undecidability. The word problem (does a given word equal $e$ ?), conjugacy problem, and isomorphism problem are all undecidable for general finitely presented groups (Novikov–Boone, Adyan–Rabin). Thus there is no algorithm that, given two presentations, decides whether they present isomorphic groups — even deciding whether a presented group is trivial is undecidable. Open and active areas include: the Andrews–Curtis conjecture (can every balanced presentation of the trivial group be reduced to the empty one by elementary moves and stabilizations? — widely believed false but unresolved); the precise classification of which groups are hyperbolic or automatic; and the profinite rigidity question of which finitely presented groups are determined by their finite quotients. The geometry of "random" presentations (Gromov's random groups) remains a fertile research frontier.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also