group in nLab

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#### Category theory
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#### Group Theory
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[[!include group theory - contents]]
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#### Monoid theory
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[[!include monoid theory - contents]]
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## 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 element|inverse]]. When written with a view toward [[group objects]] (see *[Internalization](#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 object|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 object|pointed]] groupoids with [[pointed functor|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 $1$-[[1-category|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 
 {#Internalization}

A **[[group object]]** (see there for more) [[internalization|internal to]] a [[category]] $C$ with [[finite products]] (hence in particular with a [[terminal object]] $\ast$) is an object $G$ together with [[morphisms]] $mult \colon G \times G\to G$, $id \colon \ast \to G$, and $inv \colon G\to G$ such that various diagrams commute expressing associativity, unitality, and inverses (see [there](group+object#InCartesianMonoidalCategory)).

Equivalently, it is a functor $C^{op}\to Grp$ whose underlying functor $C^{op} \to Set$ is [[representable functor|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 category theory|higher categorical]] and [[homotopy theory|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-groupoid]]s

* 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 [[groupoid object in an (infinity,1)-category|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 [[groupoid object in an (infinity,1)-category|group object in an (infinity,1)-category]] is [[delooping|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 [[loop (algebra)|loops]], then remove the identity to get [[quasigroups]];
* or remove all of the above to get [[magma|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 [[semicategory|semicategories]]
* etc.

## Examples

### Special types and classes

* [[simple group]]

* [[finite group]], [[progroup]]

  * [[classification of finite simple groups]]

  * [[sporadic finite simple groups]]

* [[abelian group]]

  * [[finite abelian group]]

* [[divisible group]]

* [[acyclic group]]

* [[topological group]]

  * [[discrete group]]

  * [[Kac-Moody group]]

* [[Lie group]]

* [[group of Lie type]]

### 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 number]]s without 0 $\mathbb{R}\setminus \{0\}$ under [[multiplication]].

* [[Prüfer group]]

Standard examples of [[Lie groups]] include the

* [[orthogonal group]]

* [[unitary group]]

* [[Spin group]], [[spin^c group]]

Standard examples of [[topological groups]] include

* [[string group]]

### Counterexamples

For more see *[[counterexamples in algebra]]*.

1. A non-[[abelian group|abelian]] [[group]], all of whose [[subgroup]]s are [[normal subgroup|normal]]:

   $$
   Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A [[finitely presented group|finitely presented]], infinite, [[simple group]]

   [[Thomson's group]] T.

1. A [[group]] that is not the [[fundamental group]] of any [[3-manifold]].

   $$
   \mathbb{Z}^4
   $$

1. Two [[finite group|finite]] non-[[isomorphism|isomorphic]] groups with the same [[order profile]].

   $$
   C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A counterexample to the converse of [[Lagrange's theorem]].

   The [[alternating group]] $A_4$ has order $12$ but no [[subgroup]] of order $6$.

1. A [[finite group]] in which the product of two [[commutator]]s 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]]

    * [[order of a group]]
  
      * [[p-primary group]]

    * [[finite group]], [[profinite group]]

    * [[finitely generated group]]

  * [[subgroup]]

    * [[torsion subgroup]]

    * [[stabilizer]],  [[centralizer]], [[normalizer]]

  * [[isogeny]]

  * [[coset]], [[coset space]]

  * [[abelian group]], [[anabelian group]], 

    * [[group completion]]

  * [[hyperbolic group]]

  * [[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]]

[[!include oidification - table]]

## Literature

For more see the references at *[[group theory]]*.

Textbook accounts:

* [[Mark A. Armstrong]]: *Groups and Symmetry*, Undergraduate Texts in Mathematics, Springer (1988) &lbrack;[doi:10.1007/978-1-4757-4034-9](https://doi.org/10.1007/978-1-4757-4034-9), [pdf](https://superoles.wordpress.com/wp-content/uploads/2014/10/lluvia.pdf)&rbrack;


The terminology "group" was introduced (for what today would more specifically be called *[[permutation groups]]*) in 

* [[Évariste Galois]], *[[Galois' last letter|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](http://dx.doi.org/10.1007/bf01443599))

Introduction of group theory into ([[quantum physics|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) &lbrack;[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)&rbrack;



Textbook account in relation to applications in [[physics]]:

* {#Sternberg94} [[Shlomo Sternberg]], *Group Theory and Physics*, Cambridge University Press 1994 ([ISBN:9780521558853](https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853))


See also:

* Wikipedia, *<a href="https://en.wikipedia.org/wiki/Group_(mathematics)">Group_(mathematics)</a>*

* [[bananaspace]], *[群](https://www.bananaspace.org/wiki/%E7%BE%A4)* (Chinese)


{#TTFormalizations} Formalization of group structure in [[dependent type theory]]:

in [[Rocq]]:

* Farida Kachapova, *Formalizing groups in type theory* &lbrack;[arXiv:2102.09125](https://arxiv.org/abs/2102.09125)&rbrack;

and with the [[univalence axiom]]

* [[unimath]] -> [UniMath.Algebra.Groups](https://unimath.github.io/doc/UniMath/d4de26f//UniMath.Algebra.Groups.html)

in [[Agda]]:

* [agda-unimath](https://unimath.github.io/agda-unimath/) -> [group-theory.groups](https://unimath.github.io/agda-unimath/group-theory.groups.html)

* [[Martín Escardó]], *[Groups](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#groups-sip)*, §3.33.10 in: *Introduction to Univalent Foundations of Mathematics with Agda* &lbrack;[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)&rbrack;


in [[cubical Agda]]:

* [[1lab]]: *[Algebra.Group](https://1lab.dev/Algebra.Group.html)*


in [[Lean]]:

* [Lean Community](https://leanprover-community.github.io/) --> [mathlib](https://leanprover-community.github.io/mathlib-overview.html) --> [algebra.group.defs](https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#top) --> [group](https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#group)

Exposition in a context of [[homotopy type theory]]:

* [[Egbert Rijke]], Section 19 in: *Introduction to Homotopy Type Theory*, Cambridge Studies in Advanced Mathematics, Cambridge University Press &lbrack;[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)&rbrack;


Alternative discussion (under [[looping and delooping]]) of groups in [[homotopy type theory]] as pointed connected [[homotopy 1-types]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 4 of: *[[Symmetry]]* (2021) &lbrack;[pdf](https://unimath.github.io/SymmetryBook/book.pdf)&rbrack;


[[!redirects groups]]

category: group theory