nLab group

Contents

Context

Category theory

Group Theory

Monoid theory

Definition
Delooping
Generalizations

Internalization
In higher categorical and homotopical contexts
Weakened axioms

Examples

Special types and classes
Concrete examples
Counterexamples

Related concepts
Literature

Definition

A group is an algebraic structure consisting of a set $G$ and a binary operation $\star$ that satisfies the group axioms, being:

associativity: $\forall a,b,c \in G: (a \star b) \star c = a \star (b \star c)$
identity: $\exists e \in G, \forall a \in G: e \star a = a \star e = a$
inverse: $\forall a \in G, \exists a^{-1} \in G: a \star a^{-1} = a^{-1} \star a = e$

It follows that the inverse $a^{-1}$ is unique for all $a$ and $G$ is non-empty.

In a broader sense, a group is a monoid in which every element has a (necessarily unique) inverse. When written with a view toward group objects (see Internalization below), one should rather say that a group is a monoid together with an inversion operation.

An abelian group is a group where the order in which two elements are multiplied is irrelevant. That is, it satisfies commutativity: $\forall a,b \in G : a \star b = b \star a$ .

Delooping

To some extent, a group “is” a groupoid with a single object, or more precisely a pointed groupoid with a single object.

The delooping of a group $G$ is a groupoid $\mathbf{B} G$ with

$Obj(\mathbf{B}G) = \{\bullet\}$
$Hom_{\mathbf{B}G}(\bullet, \bullet) = G$ .

Since for $G, H$ two groups, functors $\mathbf{B}G \to \mathbf{B}H$ are canonically in bijection with group homomorphisms $G \to H$ , this gives rise to the following statement:

Let Grpd be the 1-category whose objects are groupoids and whose morphisms are functors (discarding the natural transformations). Let Grp be the category of groups. Then the delooping functor

\mathbf{B} \colon Grp \to Grpd

is a full and faithful functor. In terms of this functor we may regard groups as the full subcategory of groupoids on groupoids with a single object.

It is in this sense that a group really is a groupoid with a single object.

But notice that it is unnatural to think of Grpd as a 1-category. It is really a 2-category, namely the sub-2-category of Cat on groupoids.

And the category of groups is not equivalent to the full sub-2-category of the 2-category of groupoids on one-object groupoids.

The reason is that two functors:

\mathbf{B}f_1, \mathbf{B}f_2 \colon \mathbf{B}G \to \mathbf{B}H

coming from two group homomorphisms $f_1, f_2 \colon G \to H$ are related by a natural transformation $\eta_h \colon \mathbf{B}f_1 \to \mathbf{B}f_2$ with single component $\eta_h \colon \bullet \mapsto h \in Mor(\mathbf{B} H)$ for each element $h \in H$ such that the homomorphisms $f_1$ and $f_2$ differ by the inner automorphism $Ad_h \colon H \to H$

(\eta_h \colon \mathbf{B}f_1 \to \mathbf{B}f_2) \Leftrightarrow (f_2 = Ad_h \circ f_1) \,.

To fix this, look at the category of pointed groupoids with pointed functors? and pointed natural transformations. Between group homomorphisms as above, only identity transformations are pointed, so $Grp$ becomes a full sub- $2$ -category of $Grpd_*$ (one that happens to be a $1$ -category). (Details may be found in the appendix to Lectures on n-Categories and Cohomology and should probably be added to pointed functor? and maybe also k-tuply monoidal n-category.)

Generalizations

Internalization

A group object internal to a category $C$ with finite products is an object $G$ together with maps $mult:G\times G\to G$ , $id:1\to G$ , and $inv:G\to G$ such that various diagrams expressing associativity, unitality, and inverses commute.

Equivalently, it is a functor $C^{op}\to Grp$ whose underlying functor $C^{op} \to Set$ is representable.

For example, a group object in Diff is a Lie group. A group object in Top is a topological group. A group object in Sch/S (the category or relative schemes) is an $S$ -group scheme. And a group object in $CAlg^{op}$ , where CAlg is the category of commutative algebras, is a (commutative) Hopf algebra.

A group object in Grp is the same thing as an abelian group (see Eckmann-Hilton argument), and a group object in Cat is the same thing as an internal category in Grp, both being equivalent to the notion of crossed module.

In higher categorical and homotopical contexts

Internalizing the notion of group in higher categorical and homotopical contexts yields various generalized notions. For instance

a 2-group is a group object in Grpd
an n-group is a group object internal to n-groupoids
an ∞-group is a group object in an (∞,1)-category.
a loop space is a group object in Top
generally there is a notion of group object in an (infinity,1)-category.

And the notion of loop space object and delooping makes sense (at least) in any (infinity,1)-category.

Notice that the relation between group objects and deloopable objects becomes more subtle as one generalizes this way. For instance not every group object in an (infinity,1)-category is deloopable. But every group object in an (infinity,1)-topos is.

Weakened axioms

Following the practice of centipede mathematics, we can remove certain properties from the definition of group and see what we get:

remove inverses to get monoids, then remove the identity to get semigroups;
or remove associativity to get loops, then remove the identity to get quasigroups;
or remove all of the above to get magmas;
or instead allow (in a certain way) for the binary operation to be partial to get groupoids, then remove inverses to get categories, and then remove identities to get semicategories
etc.

Examples

Special types and classes

Concrete examples

Standard examples of finite groups include the

group of order 2 $\;\mathbb{Z}/2\mathbb{Z}$
symmetric group $\;\Sigma_n$
cyclic group
braid group $Br_n$
Monster group

Standard examples of non-finite groups include thr

group of integers $\mathbb{Z}$ (under addition);
group of real numbers without 0 $\mathbb{R}\setminus \{0\}$ under multiplication.
Prüfer group

Standard examples of Lie groups include the

Standard examples of topological groups include

string group

Counterexamples

For more see counterexamples in algebra.

A non-abelian group, all of whose subgroups are normal:

$Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle$
A finitely presented, infinite, simple group

Thomson's group T.
A group that is not the fundamental group of any 3-manifold.

$\mathbb{Z}^4$
Two finite non-isomorphic groups with the same order profile.

$C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle$
A counterexample to the converse of Lagrange's theorem.

The alternating group $A_4$ has order $12$ but no subgroup of order $6$ .
A finite group in which the product of two commutators is not a commutator.

$G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16}$

Related concepts

2-group
n-group
∞-group
monoid, monoid object,
group, group object
- discrete group
- subgroup
  - torsion subgroup
  - stabilizer, centralizer, normalizer
- isogeny
- coset, coset space
- abelian group, anabelian group,
  - group completion
- group commutator, commutator subgroup, abelianization
- group character
- group cohomology
- group extension
- normed group, bornological group
- topological group, Lie group,
- loop group
- cogroup
- multivalued group
- is a commutative pregroup as mentioned in pregroup grammar
ring, ring object
automorphism group, automorphism 2-group, automorphism ∞-group,
- group of bisections
center, center of an ∞-group,
inner automorphism group
outer automorphism group, outer automorphism ∞-group
group presentation
groupal setoid

algebraic structure	oidification
magma	magmoid
pointed magma with an endofunction	setoid/Bishop set
unital magma	unital magmoid
quasigroup	quasigroupoid
loop	loopoid
semigroup	semicategory
monoid	category
anti-involutive monoid	dagger category
associative quasigroup	associative quasigroupoid
group	groupoid
flexible magma	flexible magmoid
alternative magma	alternative magmoid
absorption monoid	absorption category
cancellative monoid	cancellative category
rig	CMon-enriched category
nonunital ring	Ab-enriched semicategory
nonassociative ring	Ab-enriched unital magmoid
ring	ringoid
nonassociative algebra	linear magmoid
nonassociative unital algebra	unital linear magmoid
nonunital algebra	linear semicategory
associative unital algebra	linear category
C-star algebra	C-star category
differential algebra	differential algebroid
flexible algebra	flexible linear magmoid
alternative algebra	alternative linear magmoid
Lie algebra	Lie algebroid
monoidal poset	2-poset
strict monoidal groupoid?	strict (2,1)-category
strict 2-group	strict 2-groupoid
strict monoidal category	strict 2-category
monoidal groupoid	(2,1)-category
2-group	2-groupoid/bigroupoid
monoidal category	2-category/bicategory

Literature

For more see also the references at group theory.

The terminology “group” was introduced (for what today would more specifically be called permutation groups) in

Évariste Galois, letter to Auguste Chevallier, (May 1832)

The original article that gives a definition equivalent to the modern definition of a group:

Heinrich Weber, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist, Mathematische Annalen 20:3 (1882), 301–329 (doi:10.1007/bf01443599)

Introduction of group theory into (quantum) physics (cf. Gruppenpest):

Hermann Weyl, §III in: Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics, Dover (1950) [ISBN:0486602699, ark:/13960/t1kh1w36w]

Textbook account in relation to applications in physics:

Shlomo Sternberg, Group Theory and Physics, Cambridge University Press 1994 (ISBN:9780521558853)