01.02.20 · foundations / groups

Free group, free product, group presentation

shipped3 tiersLean: none

Anchor (Master): Dyck 1882 Math. Ann. 20; Nielsen 1921 Math. Ann. 94; Schreier 1927 Abh. Hamburg 5; Magnus-Karrass-Solitar Combinatorial Group Theory; Serre Trees 1980

Intuition [Beginner]

Imagine you have an alphabet of letters, say $a$ and $b$ , and you form words by stringing them together: $ab$ , $ba$ , $aab$ , and so on. You can also use inverses $a^{- 1}$ and $b^{- 1}$ , and you cancel whenever a letter meets its inverse. With no further rules, this system of words is a free group. The word "free" means free of any extra relations beyond the minimum needed to have a group.

A group presentation adds rules. You start with a free group, then declare certain words equal to the identity. Requiring $a^{3} = e$ and $b^{2} = e$ adds constraints that reduce the set of distinct words. The presentation $⟨ a, b ∣ a^{3} = e, b^{2} = e, bab = a^{- 1} ⟩$ produces a group with exactly $6$ elements, the symmetries of a triangle.

Why does this concept exist? Free groups and presentations give a universal way to describe any group: every group is a quotient of a free group, so presentations encode group structure as generators plus relations.

Visual [Beginner]

The diagram shows the Cayley graph of the free group $F_{2}$ on two generators $a$ and $b$ . Each vertex is a group element (a reduced word), and edges labeled $a$ and $b$ connect each word to its extensions. The graph is an infinite tree with four branches at every vertex — no cycles, because there are no relations to create loops.

Adding a relation like $a^{2} = e$ would collapse pairs of vertices and introduce cycles, breaking the tree structure.

Worked example [Beginner]

Consider the dihedral group $D_{3}$ (symmetries of an equilateral triangle) given by the presentation $⟨ r, s ∣ r^{3} = e, s^{2} = e, sr s = r^{- 1} ⟩$ .

Step 1. The generators are $r$ (rotation by $120$ degrees) and $s$ (a reflection). The relations $r^{3} = e$ and $s^{2} = e$ bound the powers of each generator. The third relation $sr s = r^{- 1}$ lets us move $s$ past $r$ at the cost of inverting $r$ : applying it gives $sr = r^{- 1} s$ and $s r^{2} = r^{- 2} s$ .

Step 2. Every word in $r$ and $s$ reduces to the form $r^{k}$ or $s r^{k}$ for $k = 0, 1, 2$ , using $s^{2} = e$ to eliminate adjacent reflections and $s r^{k} = r^{- k} s$ to push all reflections to the left.

Step 3. The six distinct elements are $e$ , $r$ , $r^{2}$ , $s$ , $sr$ , $s r^{2}$ . Any other word collapses into one of these by the reduction rules.

What this tells us: two generators and three relations capture all six symmetries of the triangle. The presentation is a compact encoding of the entire group.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Free group, universal property). Let $S$ be a set. The free group $F (S)$ on $S$ is a group equipped with a map $ι : S \to F (S)$ satisfying the following universal property: for every group $G$ and every set map $f : S \to G$ , there exists a unique group homomorphism $\overset{ˉ}{f} : F (S) \to G$ such that $\overset{ˉ}{f} \circ ι = f$ .

The free group admits a concrete construction. Let $S^{- 1} = {s^{- 1} : s \in S}$ be a disjoint copy of $S$ . A word in $S$ is a finite sequence $w = x_{1} x_{2} \dots x_{n}$ with each $x_{i} \in S \cup S^{- 1}$ . A word is reduced if no adjacent pair has the form $s \cdot s^{- 1}$ or $s^{- 1} \cdot s$ . Every word reduces to a unique reduced word by iteratively deleting cancelling pairs. The elements of $F (S)$ are reduced words, with multiplication given by concatenation followed by reduction to canonical form ^{[Dummit-Foote §6.3]}.

When $S = {s_{1}, \dots, s_{n}}$ , write $F_{n} = F (S)$ and call $n$ the rank. The group $F_{0}$ is the identity group; $F_{1} ≅ Z$ .

Definition (Group presentation). A group presentation $⟨ S ∣ R ⟩$ specifies a set $S$ of generators and a set $R$ of words in $F (S)$ called relators. The group it defines is the quotient $F (S) / N (R)$ , where $N (R)$ is the smallest normal subgroup of $F (S)$ containing $R$ .

Definition (Free product). Let $G$ and $H$ be groups. The free product $G * H$ is the group satisfying the universal property: for every group $K$ and every pair of homomorphisms $φ_{G} : G \to K$ and $φ_{H} : H \to K$ , there exists a unique homomorphism $φ : G * H \to K$ extending both. Concretely, elements of $G * H$ are reduced words $g_{1} h_{1} g_{2} h_{2} \dots$ with alternating factors from $G ∖ {e}$ and $H ∖ {e}$ , multiplied by concatenation and cancellation.

Counterexamples to common slips [Intermediate+]

Free group vs. free abelian group. The free group $F (S)$ is not abelian when $∣ S ∣ \geq 2$ . The free abelian group on $S$ is the quotient $F (S) / [F (S), F (S)] ≅ Z^{∣ S ∣}$ , which imposes commutativity that $F (S)$ does not have.
Presentations can be deceptive. The presentations $⟨ a ∣ ⟩$ and $⟨ a ∣ a^{6} = e ⟩$ define groups of different orders (infinite vs. $6$ ), but it is undecidable in general whether two presentations define isomorphic groups (Novikov 1955, Boone 1958).
Subgroup rank can exceed the ambient rank. Every subgroup of a free group is free (Nielsen-Schreier), but a subgroup of $F_{2}$ can have rank larger than $2$ . A subgroup of index $3$ in $F_{2}$ is free of rank $4$ .

Key theorem with proof [Intermediate+]

Theorem (Existence of presentations). Every group $G$ is isomorphic to a quotient $F (S) / N$ of a free group. Equivalently, every group admits a presentation.

Proof. Let $S$ be any generating set for $G$ (for instance, $S = G$ ). The set inclusion $f : S ↪ G$ extends by the universal property to a unique homomorphism $π : F (S) \to G$ .

The map $π$ is surjective. Every element $g \in G$ is a product $g = s_{1}^{ϵ_{1}} s_{2}^{ϵ_{2}} \dots s_{k}^{ϵ_{k}}$ with $s_{i} \in S$ and $ϵ_{i} \in {+ 1, - 1}$ , since $S$ generates $G$ . The corresponding word $s_{1}^{ϵ_{1}} s_{2}^{ϵ_{2}} \dots s_{k}^{ϵ_{k}}$ in $F (S)$ maps to $g$ under $π$ .

By the first isomorphism theorem, $F (S) / ker (π) ≅ G$ . Setting $N = ker (π)$ , which is normal in $F (S)$ , the quotient $F (S) / N$ is isomorphic to $G$ .

A presentation for $G$ is $⟨ S ∣ R ⟩$ where $R$ is any set of words whose normal closure $N (R)$ equals $N = ker (π)$ . A word $w \in F (S)$ belongs to $ker (π)$ if and only if the corresponding product in $G$ equals the identity. $□$

Bridge. This theorem is the foundational reason that group presentations are universal: this is exactly the statement that any group can be described by generators and relations. The central insight is that the kernel of the canonical surjection captures all internal constraints of the group as relators, and this identifies the study of arbitrary groups with the study of quotients of free groups. The bridge is between combinatorial data (a list of generators and relators) and algebraic structure (the group up to isomorphism). This result builds toward 01.02.02 where the isomorphism theorems supply the quotient machinery, and appears again in 01.02.19 where the tensor algebra satisfies an analogous universal property for modules over a ring, with the free group construction as the prototypical free-forgetful adjunction.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Prove that the free product $Z /2 Z * Z /2 Z$ is an infinite group.

Hint

Use the normal form for free products. Elements are alternating words in the two factors. Show that no relation can reduce all words to finitely many equivalence classes.

Answer

Let $G = ⟨ a ∣ a^{2} = e ⟩ ≅ Z /2 Z$ and $H = ⟨ b ∣ b^{2} = e ⟩ ≅ Z /2 Z$ . In the free product $G * H$ , every element has a unique reduced form as an alternating word in $a$ and $b$ (since the only non-identity elements of $G$ and $H$ are $a$ and $b$ respectively). The reduced words are $e, a, b, ab, ba, aba, bab, abab, baba, \dots$ No two consecutive letters can be the same (since $a^{2} = e$ and $b^{2} = e$ cause immediate reduction). The words $ab, abab, ababab, \dots, (ab)^{n}$ are all distinct for different $n$ , because the normal form theorem for free products guarantees that two reduced words represent the same element only if they are identical. Since there are infinitely many such words, the group is infinite. This group is the infinite dihedral group $D_{\infty}$ .

Exercise 5 (medium, symbolic).

Prove that the universal property characterises $F (S)$ up to unique isomorphism. That is, if $F_{1}$ and $F_{2}$ are both groups with maps $ι_{1} : S \to F_{1}$ and $ι_{2} : S \to F_{2}$ satisfying the universal property of the free group on $S$ , then $F_{1} ≅ F_{2}$ via a unique isomorphism compatible with the inclusion maps.

Hint

Apply the universal property of $F_{1}$ to the map $ι_{2} : S \to F_{2}$ , and then the universal property of $F_{2}$ to the map $ι_{1} : S \to F_{1}$ . Show that the resulting maps are mutually inverse.

Answer

By the universal property of $F_{1}$ applied to the map $ι_{2} : S \to F_{2}$ , there exists a unique homomorphism $φ : F_{1} \to F_{2}$ with $φ \circ ι_{1} = ι_{2}$ . By the universal property of $F_{2}$ applied to the map $ι_{1} : S \to F_{1}$ , there exists a unique homomorphism $ψ : F_{2} \to F_{1}$ with $ψ \circ ι_{2} = ι_{1}$ .

Consider $ψ \circ φ : F_{1} \to F_{1}$ . We have $(ψ \circ φ) \circ ι_{1} = ψ \circ ι_{2} = ι_{1}$ . The identity $id_{F_{1}}$ also satisfies $id_{F_{1}} \circ ι_{1} = ι_{1}$ . By the uniqueness clause of the universal property applied to $ι_{1} : S \to F_{1}$ , we conclude $ψ \circ φ = id_{F_{1}}$ . Symmetrically, $φ \circ ψ = id_{F_{2}}$ . Therefore $φ$ and $ψ$ are mutually inverse isomorphisms.

Exercise 8 (hard, symbolic).

Prove that a free group is torsion-free: if $w \in F (S)$ is a reduced word different from the identity and $k \geq 1$ , then $w^{k} \neq = e$ in $F (S)$ .

Hint

Write $w = x_{1} x_{2} \dots x_{n}$ in reduced form. Analyse what happens when you concatenate $w$ with itself. Consider whether the last letter of $w$ is the inverse of the first letter, and show that cancellation is bounded.

Answer

Let $w = x_{1} x_{2} \dots x_{n}$ be a reduced word with $n \geq 1$ . Write $w = uv$ where $v$ is the longest suffix of $w$ that cancels with the beginning of $w$ when forming $w^{2}$ . Specifically, if $x_{n} \neq = x_{1}^{- 1}$ , then no cancellation occurs and $w^{2} = x_{1} \dots x_{n} x_{1} \dots x_{n}$ is reduced and non-empty. If $x_{n} = x_{1}^{- 1}$ , there is some cancellation but it is bounded: the cancellation between two consecutive copies of $w$ can reduce at most $2 n - 2$ letters from the combined $2 n$ letters, leaving at least $2$ letters.

Formally, write $w = u v^{- 1}$ where $v$ is the maximal initial segment such that $w$ ends with $v^{- 1}$ . Then $w = u v^{- 1}$ and $w$ also begins with $v$ , so $w^{2} = u v^{- 1} u v^{- 1}$ . Since $w$ is reduced and $v$ is the longest cancelling prefix-suffix pair, the word $v^{- 1} u$ is non-empty (and cyclically reduced). Thus $w^{k} = u (v^{- 1} u)^{k - 1} v^{- 1}$ , which is non-empty for all $k \geq 1$ because $v^{- 1} u$ is a reduced word that does not cancel against itself. Therefore $w^{k} \neq = e$ for all $k \geq 1$ .

Advanced results [Master]

Theorem 1 (Nielsen-Schreier). Every subgroup of a free group is free. If $H \leq F_{r}$ has index $n < \infty$ , then $H ≅ F_{s}$ where $s = n (r - 1) + 1$ (the Schreier index formula) ^{[Schreier 1927]}. The formula shows that a subgroup of finite index in $F_{r}$ has strictly larger rank whenever $r \geq 2$ and $n \geq 2$ .

Theorem 2 (Normal form for free products). Let $G * H$ be the free product. Every element has a unique expression as $g_{1} h_{1} g_{2} h_{2} \dots g_{k} h_{k}$ or $h_{1} g_{1} h_{2} g_{2} \dots h_{k} g_{k}$ with $g_{i} \in G ∖ {e}$ , $h_{i} \in H ∖ {e}$ , and $k \geq 1$ ; or as the identity. This normal form theorem implies $G * H$ is infinite whenever both $G$ and $H$ are non-identity groups.

Theorem 3 (Grushko-Neumann). The rank of the free product satisfies $rank (G * H) = rank (G) + rank (H)$ when both groups are free. This means free product is additive on rank, analogous to dimension additivity for direct sums of vector spaces.

Theorem 4 (Kurosh subgroup theorem). Every subgroup of a free product $G_{1} * G_{2} * \dots * G_{n}$ is itself a free product of a free group with conjugates of subgroups of the factors $G_{i}$ . This generalises Nielsen-Schreier: when each $G_{i} = Z$ , the free product is a free group, and the theorem recovers the statement that subgroups of free groups are free ^{[Magnus-Karrass-Solitar]}.

Theorem 5 (Tietze transformations). Two finite presentations define isomorphic groups if and only if one can be obtained from the other by a finite sequence of Tietze transformations: adding a redundant relator, removing a relator that follows from others, adding a new generator together with a relation defining it, or removing a generator that can be expressed in terms of the others. Tietze transformations give a computational method for manipulating presentations ^{[Lyndon-Schupp]}.

Theorem 6 (Word problem, undecidability). There exists a finitely presented group for which the word problem is undecidable: no algorithm can determine, given an arbitrary word $w$ in the generators, whether $w$ equals the identity in the group (Novikov 1955 Doklady Akad. Nauk SSSR 107; Boone 1958 Ann. Math. 68). The uniform word problem — deciding, given an arbitrary presentation and a word, whether the word is the identity — is also undecidable.

Theorem 7 (Free product with amalgamation). Let $G$ and $H$ be groups with a common subgroup $A \leq G$ and $A \leq H$ via embeddings $φ_{G}$ and $φ_{H}$ . The free product with amalgamation $G *_{A} H$ is the quotient $G * H / N$ where $N$ is the normal closure of ${φ_{G} (a) φ_{H} (a)^{- 1} : a \in A}$ . The universal property: a homomorphism from $G *_{A} H$ to $K$ is a pair of homomorphisms $G \to K$ and $H \to K$ that agree on $A$ .

Theorem 8 (Bass-Serre theory, statement). A group $G$ acts without inversions on a tree $T$ without edge inversions if and only if $G$ is either a free group (when $G$ acts freely) or a free product with amalgamation or HNN extension (when the action has edge stabilisers). The fundamental group of a graph of groups decomposes as an iterated free product with amalgamation. Conversely, every such decomposition determines a tree on which $G$ acts, the Bass-Serre tree ^{[Serre 1980]}.

Synthesis. The free group is the foundational reason that combinatorial group theory exists as a coherent subject: this is exactly the universal construction encoding all groups as quotients of letter-strings modulo relators. The central insight is that the Nielsen-Schreier theorem identifies subgroups of free groups with free groups of computable rank, and this is dual to the Kurosh theorem which describes subgroups of free products. Putting these together with the Grushko-Neumann rank formula, the rank of a free group becomes a well-defined invariant analogous to dimension for vector spaces. The bridge is between combinatorial presentations and geometric group actions on trees via Bass-Serre theory, and the pattern generalises to free products with amalgamation and HNN extensions as the fundamental building blocks in the decomposition of groups.

Full proof set [Master]

Proposition 1 (Free groups are torsion-free, detailed). Let $w \in F (S)$ be a non-identity reduced word. Then $w^{k} \neq = e$ for all positive integers $k$ .

Proof. Write $w = x_{1} x_{2} \dots x_{n}$ in reduced form with $n \geq 1$ . Define the cyclic reductions of $w$ : iteratively remove matching first and last letters $x_{1} = x_{n}^{- 1}$ until either the word is empty or the first and last letters do not cancel. Call the result $w_{0} = c (w)$ , the cyclically reduced form of $w$ .

If $w_{0} = e$ , then $w = x_{1} \dots x_{m} x_{m}^{- 1} \dots x_{1}^{- 1}$ for some $m$ , but $w$ is reduced, so $x_{m} \neq = x_{m}^{- 1}$ , meaning $w_{0} \neq = e$ unless $w = e$ . Since $w \neq = e$ , we have $w_{0} \neq = e$ .

Write $w = a w_{0} a^{- 1}$ for some word $a$ (this is the reduction process). Then $w^{k} = a w_{0}^{k} a^{- 1}$ . Since $w_{0}$ is cyclically reduced, $w_{0}^{k}$ is reduced for all $k$ (no cancellation occurs between consecutive copies of $w_{0}$ ). Therefore $w^{k} = a w_{0}^{k} a^{- 1}$ , where $w_{0}^{k}$ is non-empty. The concatenation $a w_{0}^{k} a^{- 1}$ reduces at most $∣ a ∣$ letters from each end, but $w_{0}^{k}$ has $k ∣ w_{0} ∣ \geq k \geq 1$ letters, so the result is non-empty for all $k \geq 1$ . $□$

Proposition 2 (Schreier index formula). Let $F_{r}$ be the free group of rank $r$ and $H \leq F_{r}$ a subgroup of finite index $n$ . Then $H$ is free of rank $s = n (r - 1) + 1$ .

Proof (sketch). Choose a Schreier transversal $T$ for $H$ in $F_{r}$ : a set of right coset representatives such that every initial segment of a representative is again a representative. For each $t \in T$ and each generator $x \in {s_{1}, \dots, s_{r}, s_{1}^{- 1}, \dots, s_{r}^{- 1}}$ , define $γ (t, x) = t x \overline{t x}^{- 1}$ , where $\overline{g}$ denotes the unique representative of $H g$ . The element $γ (t, x)$ lies in $H$ .

The Schreier generators ${γ (t, s_{i}) : t \in T, 1 \leq i \leq r} ∖ {e}$ form a free generating set for $H$ . To count them, observe: there are $n \cdot r$ pairs $(t, s_{i})$ with $t \in T$ and $1 \leq i \leq r$ . Among these, exactly $n - 1$ give $γ (t, s_{i}) = e$ (one for each non-identity coset, since for the representative $t$ of the identity coset, $γ (e, s_{i}) = s_{i} \overline{s_{i}}^{- 1} \neq = e$ for all $i$ ). The number of non-identity Schreier generators is $n r - (n - 1) = n (r - 1) + 1$ , giving rank $s = n (r - 1) + 1$ . $□$

Connections [Master]

Group 01.02.01. The free group on a set $S$ is the most general group that can be built from $S$ : it adds no relations beyond those forced by the group axioms. The foundational reason this matters is that every group arises as a quotient of a free group, making free groups the universal building blocks in the category of groups. The bridge is between the categorical universal property of free objects and the concrete word-reduction construction that makes the free group computable.
Subgroup, coset, quotient group, isomorphism theorems 01.02.02. The isomorphism theorems underpin the theory of group presentations: the first isomorphism theorem produces the presentation $G ≅ F (S) / ker (π)$ , and the correspondence between subgroups of $G$ and subgroups of $F (S)$ containing $ker (π)$ relies on the lattice isomorphism theorem. Normal subgroups of free groups correspond to relator sets, and the quotient construction from that unit appears again here as the mechanism converting a free group and a normal subgroup into an arbitrary group.
Tensor algebra, exterior algebra, symmetric algebra 01.02.19. The free group and the tensor algebra satisfy formally analogous universal properties: both are free objects in their respective categories (groups and associative algebras), both are constructed as quotients of word monoids, and both serve as the starting point for further quotient constructions (group presentations for the free group, exterior and symmetric algebras for the tensor algebra). The free functor from sets to groups is the group-theoretic parallel of the tensor algebra functor from modules to algebras.

Historical & philosophical context [Master]

Walther von Dyck 1882 introduced group presentations in Gruppentheoretische Studien (Math. Ann. 20), establishing that a group can be specified by generators and defining relations ^{[Dyck 1882]}. Dyck's work made precise the informal practice, going back to Cayley, of describing groups by symbols and rules.

Jakob Nielsen 1921 developed the theory of free groups in Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden (Math. Ann. 94), proving that the automorphism group of $F_{n}$ is generated by Nielsen transformations (analogues of elementary row operations) and that rank is an invariant of free groups ^{[Nielsen 1921]}.

Otto Schreier 1927 proved the subgroup theorem in Die Untergruppen der freien Gruppen (Abh. Hamburg 5), showing that every subgroup of a free group is free and deriving the index-rank formula ^{[Schreier 1927]}. The modern geometric viewpoint — free groups as fundamental groups of graphs, and Bass-Serre theory as the study of groups acting on trees — crystallised with Serre's Arbres, amalgames, SL2 (1977, English translation Trees 1980), which unified the algebraic theory of free products with amalgamation with the topological theory of covering spaces and group actions.

Bibliography [Master]

@article{Dyck1882,
  author = {Dyck, Walther von},
  title = {Gruppentheoretische Studien},
  journal = {Mathematische Annalen},
  volume = {20},
  year = {1882},
  pages = {1--44},
}

@article{Nielsen1921,
  author = {Nielsen, Jakob},
  title = {Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden},
  journal = {Mathematische Annalen},
  volume = {94},
  year = {1921},
  pages = {249--272},
}

@article{Schreier1927,
  author = {Schreier, Otto},
  title = {Die Untergruppen der freien Gruppen},
  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hamburg},
  volume = {5},
  year = {1927},
  pages = {161--183},
}

@article{Novikov1955,
  author = {Novikov, Petr S.},
  title = {Ob algoritmi\v{c}esko\u{i} nerazre\v{s}imosti problemy to\v{z}destva slov v teorii grupp},
  journal = {Doklady Akademii Nauk SSSR},
  volume = {107},
  year = {1955},
  pages = {954--956},
}

@article{Boone1958,
  author = {Boone, William W.},
  title = {The word problem},
  journal = {Annals of Mathematics},
  volume = {68},
  year = {1958},
  pages = {794--812},
}

@book{MagnusKarrassSolitar,
  author = {Magnus, Wilhelm and Karrass, Abraham and Solitar, Donald},
  title = {Combinatorial Group Theory},
  publisher = {Dover Publications},
  year = {2004},
  note = {Reprint of the 1976 Interscience edition},
}

@book{SerreTrees,
  author = {Serre, Jean-Pierre},
  title = {Trees},
  publisher = {Springer-Verlag},
  year = {1980},
}

@book{LyndonSchupp,
  author = {Lyndon, Roger C. and Schupp, Paul E.},
  title = {Combinatorial Group Theory},
  publisher = {Springer-Verlag},
  year = {1977},
}

@book{DummitFooteGroups,
  author = {Dummit, David S. and Foote, Richard M.},
  title = {Abstract Algebra},
  publisher = {John Wiley \& Sons},
  year = {2004},
  edition = {3rd},
}

Prerequisites

01.02.01
01.02.02

Tier anchors

beginner: building groups from letters and rules
intermediate: Dummit-Foote §6.3; Lang Algebra §I.12, §I.7
master: Dyck 1882 Math. Ann. 20; Nielsen 1921 Math. Ann. 94; Schreier 1927 Abh. Hamburg 5; Magnus-Karrass-Solitar Combinatorial Group Theory; Serre Trees 1980

References

TODO_REF
Dyck — Gruppentheoretische Studien · Math. Ann. 20, 1882
TODO_REF
Nielsen — Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden · Math. Ann. 94, 1921
TODO_REF
Schreier — Die Untergruppen der freien Gruppen · Abh. Hamburg 5, 1927
TODO_REF
Magnus-Karrass-Solitar — Combinatorial Group Theory · Dover, 2004
TODO_REF
Serre — Trees · Springer, 1980
TODO_REF
Dummit-Foote — Abstract Algebra · §6.3
TODO_REF
Lyndon-Schupp — Combinatorial Group Theory · Springer, 1977

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 40m
master: 75m