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\begin{document}

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\section*{finite group}

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

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cauchys_theorem}{Cauchy's theorem}\dotfill \pageref*{cauchys_theorem} \linebreak
\noindent\hyperlink{feitthompson_theorem}{Feit-Thompson theorem}\dotfill \pageref*{feitthompson_theorem} \linebreak
\noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{most_finite_groups_are_nilpotent}{``Most finite groups are nilpotent''}\dotfill \pageref*{most_finite_groups_are_nilpotent} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

A \textbf{finite group} is a [[group]] whose underlying [[set]] is [[finite set|finite]].

This is equivalently a [[group object]] in [[FinSet]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{cauchys_theorem}{}\subsubsection*{{Cauchy's theorem}}\label{cauchys_theorem}

Let $G$ be a finite group with [[order of a group|order]] ${\vert G\vert} \in \mathbb{N}$.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Cauchy)}

If a [[prime number]] $p$ divides ${\vert G\vert}$, then equivalently

\begin{itemize}%
\item $G$ has an element of [[order of an element|order]] $p$;


\item $G$ has a [[subgroup]] of [[order of a group]] $p$.



\end{itemize}
\end{theorem}
See at \emph{[[Cauchy's theorem]]} for more.

\hypertarget{feitthompson_theorem}{}\subsubsection*{{Feit-Thompson theorem}}\label{feitthompson_theorem}

\begin{theorem}
\label{}\hypertarget{}{}
Every finite group of odd [[order]] is a [[solvable group]].

\end{theorem}
See at \emph{[[Feit-Thompson theorem]]}.

\hypertarget{classification}{}\subsubsection*{{Classification}}\label{classification}

The structure of finite groups is a very hard problem; the classification of finite [[simple group]]s alone is one of the largest theorems ever proved (certainly if measured by number of journal pages needed for a complete proof).

All finite groups are built out of [[simple group]]s, but the ways to do this have not (yet?) been fully classified.

A point of view that can be useful in particular cases -- more useful than the [[Jordan-Hölder theorem]] -- is provided by the [[F\emph{-theorem]], due to [[Hans Fitting]] in the [[solvable group|solvable]] case and [[Helmut Bender]] in the general case. It states that $C_G(F^*(G))=Z(F^*(G))$, where $F^*(G)$ is the [[generalized Fitting subgroup]] of $G$, defined below, $C_G(F^*(G))$ is the [[subgroup]] of $G$ consisting of all elements commuting with every element of $F^*(G)$, and $Z(H)$ for any group $H$ is the [[center]] of $H$, the subgroup of $H$ consisting of all elements commuting with every element of $H$. Thus $G$ is somehow assembled from $F^*(G)$, whose structure has some easy features, and $G/C_G(F^*(G))$, which is isomorphic to a subgroup of the [[automorphism group]] of $F^*(G)$ and which has a [[quotient]] group [[isomorphic]] to $G/F^*(G)$.}

One definition of $F^*(G)$ is that it is the subgroup generated by all [[normal subgroup]]s $N$ of $G$ possessing subgroups $N_1,N_2,\dots, N_r$ for some integer $r$ such that $N=N_1N_2\cdots N_r$; $x_i x_j=x_j x_i$ for all $x_i\in N_i$, $x_j\in N_j$, and distinct subscripts $i$ and $j$; and each $N_i$ either has prime power order or is a [[quasisimple group]]. Bender proved that $F^*(G)$ itself enjoys these properties.

Finally a group $H$ is called quasisimple if and only if $H=[H,H]$ and $H/Z(H)$ is simple. The finite quasisimple groups have been classified, as a consequence of the classification of finite simple groups and the calculation of the [[Schur multiplier]] of each finite simple group.

For more on this see

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

\end{itemize}
\hypertarget{most_finite_groups_are_nilpotent}{}\paragraph*{{``Most finite groups are nilpotent''}}\label{most_finite_groups_are_nilpotent}

The meaning of the title is this curious fact (based on empirical evidence, anyway): if we are counting isomorphism classes of groups up to a given order, then most of them are $2$-[[p-primary group|primary]] groups (and therefore [[nilpotent group|nilpotent]]; see [[class equation]]). For example, it is reported that ``out of the 49,910,529,484 groups of order at most 2000, a staggering 49,487,365,422 of them have order 1024''.

(It has also been suggested that a more meaningful weighting would divide each isomorphism class representative by the order of its [[automorphism group]]; as of this writing the nLab authors don't know how much this would affect the strength of the assertion ``most finite groups are nilpotent''.)

Discussion can be found \href{http://mathoverflow.net/a/114666/2926}{here} and \href{https://golem.ph.utexas.edu/category/2012/11/almost_all_of_the_first_50_bil.html}{here}.

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

\begin{itemize}%
\item For every [[natural number]] $n \in \mathbb{N}$, the [[cyclic group]]

\begin{displaymath}
\mathbb{Z}_n := \mathbb{Z}/n \mathbb{Z}
\end{displaymath}
is finite.


\item The largest finite group that is also a [[sporadic simple group]], i.e., does not belong(up to isomorphism) to the infinite family of the alternating groups or to the infinite family of finite groups of Lie type, is the [[Monster group]].


\item [[finite subgroups of SO(3)]] and [[finite subgroups of SU(2)]] have an [[ADE classification]]


\item the [[general linear group]] over [[prime fields]] are finit, such as [[GL(2,3)]]


\item [[wallpaper group]]


\item see also at \emph{[[finite abelian group]]}.



\end{itemize}
[[!include ADE -- table]]

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

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


\item [[finite group scheme]]


\item [[Burnside ring]], [[Segal conjecture]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\begin{itemize}%
\item \emph{\href{http://brauer.maths.qmul.ac.uk/Atlas/v3/}{Atlas of finite group representations}}

\end{itemize}
Discussion of [[group characters]] and [[group cohomology]] of finite groups includes

\begin{itemize}%
\item [[Michael Atiyah]], \emph{Characters and cohomology of finite groups}, Publications Math\'e{}matiques de l'IH\'E{}S, 9 (1961), p. 23-64 (\href{http://www.numdam.org/item?id=PMIHES_1961__9__23_0}{Numdam})


\item [[Alejandro Adem]], R.James Milgram, \emph{Cohomology of Finite Groups}, Springer 2004


\item [[Narthana Epa]], \emph{Platonic 2-groups}, 2010 (\href{http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf}{pdf})



\end{itemize}
Discussion of [[free actions]] of finite groups on [[n-spheres]] (see also at \emph{[[ADE classification]]}) includes

\begin{itemize}%
\item [[John Milnor]], \emph{Groups which act on $S^n$ without fixed points}, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 (\href{http://www.jstor.org/stable/2372566}{JSTOR})


\item Adam Keenan, \emph{Which finite groups act freely on spheres?}, 2003 (\href{http://www.math.utah.edu/~keenan/actions.pdf}{pdf})



\end{itemize}
[[!redirects finite groups]]



\end{document}