Group Presentations — Mathematics Knowledge System

Overview

A presentation describes a group compactly by naming a set of generators and a set of defining relations (equations among the generators) that, together with the axioms of a group, determine the group completely: ⟨generators | relations⟩. Rather than listing every element and every product, a presentation gives a recipe — start with the free group on the generators, then quotient by the smallest normal subgroup containing the relators, so that the relations hold and nothing extra is forced. Presentations let you specify enormous or infinite groups with a handful of symbols, and they underlie combinatorial group theory, computational group theory (e.g. the Todd–Coxeter algorithm), and topology (fundamental groups of spaces are naturally presented this way).

Intuition

Think of a presentation as a set of building instructions rather than a finished blueprint: 'take these generating moves, and impose exactly these rules — nothing more.' Two elements of the presented group are equal exactly when one word can be transformed into the other using the group axioms plus the stated relations (and their consequences). This is powerful but also subtle: since a presentation only forces what the relations demand, seemingly small changes to the relator list can produce wildly different groups — even algorithmically undecidable ones (the word problem for general presentations is unsolvable), which is part of why concrete, well-understood presentations like those for Dₙ or free groups are so valued.

Formal Definition

Definition

Given a set S of generators and a set R of words in S ∪ S⁻¹ (the relators), the presentation ⟨S | R⟩ denotes the group

\langle S \mid R \rangle = F(S) / N(R)

Presented group: free group on S modulo the normal closure of R

N(R) = \bigcap \{ K \trianglelefteq F(S) : R \subseteq K \}

Normal closure — smallest normal subgroup containing R

\langle r, s \mid r^n,\ s^2,\ srsr \rangle \cong D_n

Example: the dihedral group as a presentation

\langle a \mid a^n \rangle \cong \mathbb{Z}_n

Example: the cyclic group as a presentation

Notation

Notation	Meaning
$\langle S \mid R \rangle$	Group presented by generators S and relators R
$F(S)$	The free group on the generating set S
$w =_G 1$	The relator word w equals the identity in the presented group

Properties

Finitely presented

\text{A group is finitely presented if it has a presentation } \langle S \mid R \rangle \text{ with } S, R \text{ both finite sets.}

Example: D_n = \langle r,s \mid r^n, s^2, srsr \rangle is finitely presented (2 generators, 3 relators).

Non-uniqueness of presentations

\text{The same group can arise from many different presentations; there is generally no algorithm to decide if two presentations give isomorphic groups.}

Von Dyck's theorem

\text{Any group } G \text{ satisfying the relations } R \text{ on generating set } S \text{ is a quotient of } \langle S \mid R \rangle.

Example: Every group satisfying r^n=s^2=(sr)^2=e is a quotient of the abstract presentation for D_n; when |G|=2n it equals D_n exactly.

Worked Examples

The free group on one generator a is just the infinite cyclic group generated by a; imposing the single relator a⁶ = e forces a to have order dividing 6.
$\langle a \mid a^6 \rangle$
No further relation is forced, so a has order exactly 6, and the group consists of the distinct powers of a.
$\{e, a, a^2, a^3, a^4, a^5\}$

Answer: ⟨a | a⁶⟩ ≅ ℤ₆ = {e, a, a², a³, a⁴, a⁵}, the cyclic group of order 6.

Practice Problems

Difficulty 4/10

What group is presented by ⟨a, b | a² , b², (ab)²⟩? (Hint: this forces a and b to commute.)

Difficulty 5/10

Explain why ⟨a | ⟩ (a single generator, no relators) is the infinite cyclic group ℤ, not any finite group.

Difficulty 6/10

Von Dyck's theorem says any group satisfying the relations of a presentation is a quotient of the presented group. Use this to explain why |G| ≤ 2n whenever G is generated by elements r, s satisfying rⁿ=s²=(sr)²=e.

Common Mistakes

Common Mistake

Assuming a presentation obviously determines the group's order or structure just by counting generators and relators.

Presentations can hide enormous complexity — the word problem (deciding if two words represent the same element) is undecidable in general, and seemingly small presentations can define infinite or even trivial groups unexpectedly.

Quiz

Summary

⟨S | R⟩ = F(S)/N(R): a presentation builds a group from generators S constrained by relators R, via the free group modulo the normal closure of R.
Von Dyck's theorem: any group satisfying the same relations is a quotient of the presented group, bounding its size.
Presentations can be deceptively complex — the general word problem is undecidable — but concrete cases like Dₙ = ⟨r,s | rⁿ,s²,(sr)²⟩ are fully understood.

References

WebsiteWikipedia — Presentation of a group