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\begin{document}

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\section*{group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{delooping}{Delooping}\dotfill \pageref*{delooping} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{internalization}{Internalization}\dotfill \pageref*{internalization} \linebreak
\noindent\hyperlink{in_higher_categorical_and_homotopical_contexts}{In higher categorical and homotopical contexts}\dotfill \pageref*{in_higher_categorical_and_homotopical_contexts} \linebreak
\noindent\hyperlink{weakened_axioms}{Weakened axioms}\dotfill \pageref*{weakened_axioms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{special_types_and_classes}{Special types and classes}\dotfill \pageref*{special_types_and_classes} \linebreak
\noindent\hyperlink{concrete_examples}{Concrete examples}\dotfill \pageref*{concrete_examples} \linebreak
\noindent\hyperlink{counterexamples}{Counterexamples}\dotfill \pageref*{counterexamples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{group} is a [[monoid]] in which every element has an [[inverse]] (necessarily unique).

An \textbf{[[abelian group]]} is a group in which moreover the order in which two elements are multiplied is irrelevant.

\hypertarget{delooping}{}\subsection*{{Delooping}}\label{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

\begin{itemize}%
\item $Obj(\mathbf{B}G) = \{\bullet\}$


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



\end{itemize}
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

\begin{displaymath}
\mathbf{B} : Grp \to Grpd
\end{displaymath}
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 \emph{not} equivalent to the full sub-2-category of the 2-category of groupoids on one-object groupoids.

The reason is that two functors:

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

\begin{displaymath}
(\eta_h : \mathbf{B}f_1 \to \mathbf{B}f_2)
  \Leftrightarrow 
  (f_2 = Ad_h \circ f_1)
  \,.
\end{displaymath}
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]].)

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{internalization}{}\subsubsection*{{Internalization}}\label{internalization}

A \textbf{[[group object]]} [[internalization|internal to]] a [[category]] $C$ with finite [[product|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 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]].

\hypertarget{in_higher_categorical_and_homotopical_contexts}{}\subsubsection*{{In higher categorical and homotopical contexts}}\label{in_higher_categorical_and_homotopical_contexts}

Internalizing the notion of \emph{group} in [[higher category theory|higher categorical]] and [[homotopy theory|homotopical]] contexts yields various generalized notions. For instance

\begin{itemize}%
\item a [[2-group]] is a group object in [[Grpd]]


\item an [[n-group]] is a group object internal to [[n-groupoid]]s


\item an [[∞-group]] is a [[group object in an (∞,1)-category]].


\item a [[loop space]] is a group object in [[Top]]


\item generally there is a notion of [[groupoid object in an (infinity,1)-category|group object in an (infinity,1)-category]].



\end{itemize}
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.

\hypertarget{weakened_axioms}{}\subsubsection*{{Weakened axioms}}\label{weakened_axioms}

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

\begin{itemize}%
\item remove inverses to get [[monoids]], then remove the identity to get [[semigroups]];
\item or remove associativity to get [[loop (algebra)|loops]], then remove the identity to get [[quasigroups]];
\item or remove all of the above to get [[magma|magmas]];
\item 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]]
\item etc.

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{special_types_and_classes}{}\subsubsection*{{Special types and classes}}\label{special_types_and_classes}

\begin{itemize}%
\item [[simple group]]


\item [[finite group]], [[progroup]]

\begin{itemize}%
\item [[classification of finite simple groups]]


\item [[sporadic finite simple groups]]



\end{itemize}

\item [[abelian group]]

\begin{itemize}%
\item [[finite abelian group]]

\end{itemize}

\item [[divisible group]]


\item [[acyclic group]]


\item [[topological group]]

\begin{itemize}%
\item [[discrete group]]


\item [[Kac-Moody group]]



\end{itemize}

\item [[Lie group]]


\item [[group of Lie type]]



\end{itemize}
\hypertarget{concrete_examples}{}\subsubsection*{{Concrete examples}}\label{concrete_examples}

Standard examples of [[finite group]]s include

\begin{itemize}%
\item [[group of order 2]] $\mathbb{Z}_2$


\item [[symmetric group]] $\Sigma_n$


\item [[cyclic group]]


\item [[braid group]] $Br_n$


\item [[Monster group]]



\end{itemize}
Standard examples of non-finite groups include

\begin{itemize}%
\item group of [[integer]]s $\mathbb{Z}$ (under [[addition]]);


\item group of [[real number]]s without 0 $\mathbb{R}\setminus \{0\}$ under [[multiplication]].


\item [[Prüfer group]]



\end{itemize}
Standard examples of [[Lie groups]] include

\begin{itemize}%
\item [[orthogonal group]]


\item [[unitary group]]


\item [[Spin group]], [[spin{\tt \symbol{94}}c group]]



\end{itemize}
Standard examples of [[topological group]]s include

\begin{itemize}%
\item [[string group]]

\end{itemize}
\hypertarget{counterexamples}{}\subsubsection*{{Counterexamples}}\label{counterexamples}

For more see [[counterexamples in algebra]].

\begin{enumerate}%
\item A non-[[abelian group|abelian]] [[group]], all of whose [[subgroup]]s are [[normal subgroup|normal]]:

\begin{displaymath}
Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
\end{displaymath}

\item A [[finitely presented group|finitely presented]], infinite, [[simple group]]

[[Thomson's group]] T.


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

\begin{displaymath}
\mathbb{Z}^4
\end{displaymath}

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

\begin{displaymath}
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
\end{displaymath}

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

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


\item A [[finite group]] in which the product of two [[commutator]]s is not a commutator.

\begin{displaymath}
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}
\end{displaymath}


\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[2-group]]


\item [[n-group]]


\item [[∞-group]]


\item [[monoid]], [[monoid object]],


\item \textbf{group}, [[group object]]

\begin{itemize}%
\item [[discrete group]]

\begin{itemize}%
\item [[order of a group]]

\begin{itemize}%
\item [[p-primary group]]

\end{itemize}

\item [[finite group]], [[profinite group]]



\end{itemize}

\item [[subgroup]]

\begin{itemize}%
\item [[torsion subgroup]]


\item [[stabilizer]], [[centralizer]], [[normalizer]]



\end{itemize}

\item [[isogeny]]


\item [[coset]], [[coset space]]


\item [[abelian group]], [[anabelian group]],

\begin{itemize}%
\item [[group completion]]

\end{itemize}

\item [[group commutator]], [[commutator subgroup]], [[abelianization]]


\item [[group character]]


\item [[group cohomology]]


\item [[group extension]]


\item [[normed group]], [[bornological group]]


\item [[topological group]], [[Lie group]],


\item [[loop group]]


\item [[cogroup]]



\end{itemize}

\item [[ring]], [[ring object]]


\item [[automorphism group]], [[automorphism 2-group]], [[automorphism ∞-group]],

\begin{itemize}%
\item [[group of bisections]]

\end{itemize}

\item [[center]], [[center of an ∞-group]],


\item [[inner automorphism group]]


\item [[outer automorphism group]], [[outer automorphism ∞-group]]


\item [[group presentation]]



\end{itemize}
[[!redirects groups]] category: group theory



\end{document}