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\begin{document}

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\section*{Monster group}

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

\hypertarget{exceptional_structures}{}\paragraph*{{Exceptional structures}}\label{exceptional_structures}

[[!include exceptional structures -- contents]]

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

[[!include group theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{presentation}{Presentation}\dotfill \pageref*{presentation} \linebreak
\noindent\hyperlink{via_coxeter_groups}{Via Coxeter groups}\dotfill \pageref*{via_coxeter_groups} \linebreak
\noindent\hyperlink{ViaAutomorphisms}{Via automorphisms of a super vertex operator algebra}\dotfill \pageref*{ViaAutomorphisms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{Monster group} $M$ is a [[finite group]] that is the largest of the [[sporadic finite simple group]]s. It has [[order of a group|order]]

\begin{displaymath}
\begin{aligned}
& 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71 
  \\
 & = 808017424794512875886459904961710757005754368000000000
 \end{aligned}
\end{displaymath}
and contains all but six (the `[[pariah groups]]') of the other 25 [[sporadic finite simple groups]] as [[subquotients]], called the \emph{[[Happy Family]]}.

See also [[Moonshine]].

\hypertarget{history}{}\subsection*{{History}}\label{history}

The Monster group was predicted to exist by [[Bernd Fischer]] and [[Robert Griess]] in 1973, as a [[simple group]] containing the [[Fischer groups]] and some other sporadic simple groups as [[subquotients]]. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of $M$ and discovered other properties and subgroups, assuming that it existed. In a famous paper

\begin{itemize}%
\item [[Robert Griess]], \emph{The Friendly Giant} , Inventiones (1982)

\end{itemize}
Griess proved the existence of the largest simple sporadic group. The author constructs ``by hand'' a non-associative but commutative algebra of dimension 196883, and showed that the [[automorphism group]] of this algebra is the conjectured friendly giant/monster simple group. The name ``Friendly Giant'' for the Monster did not take on.

After Griess found this algebra [[Igor Frenkel]], [[James Lepowsky]] and Meurman and/or Borcherds showed that the Griess algebra is just the degree 2 part of the infinite dimensional [[Moonshine vertex algebra]].

There is a school of thought, going back to at least [[Israel Gelfand]], that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see \href{http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html}{this nCafe post}). One version of this, in the case of the Monster (and perhaps for other sporadic groups via [[Moonshine]] phenomena) is that what we know as the Monster is just a shadow of a [[2-group]], as the Monster can be constructed as an automorphism group of a [[conformal field theory]], a structure rich enough to have a automorphism 2-group(oid) (see \href{http://golem.ph.utexas.edu/category/2008/10/john_mckay_visits_kent.html#c019440}{this nCafe discussion}).

\hypertarget{presentation}{}\subsection*{{Presentation}}\label{presentation}

\hypertarget{via_coxeter_groups}{}\subsubsection*{{Via Coxeter groups}}\label{via_coxeter_groups}

The Monster admits a reasonably succinct description in terms of [[Coxeter groups]]. Let $[n]$ denote the linear [[graph]] with vertices $0, 1, \ldots, n$ with an edge between adjacent numbers $i, i+1$ and no others. If $1$ is the terminal (1-element) graph, there is a map $0: 1 \to [n]$, mapping the vertex of $1$ to the vertex $0$. Regarding this as an object in the [[undercategory]] $1 \downarrow Graph$, let $Y_{443}$ be the [[coproduct]] of the three objects $0: 1 \to [4]$, $0: 1 \to [4]$, $0: 1 \to [3]$ in $1 \downarrow Graph$. This (pointed) graph has 12 elements and is shaped like a $Y$, with arms of length 4, 4, 3 emanating from a central vertex of valence $3$.

Regard $Y_{443}$ as a [[Coxeter diagram]]. The associated [[Coxeter group]] $C_{443}$ is given by a [[group presentation]] with 12 generators (represented by the vertices) of order $2$ (so 12 relators of the form $x^2 = 1$), with a relation $(x y)^3 = 1$ if $x, y$ are adjacent vertices (so 11 relators, one for each edge), and $x y = y x$ if $x, y$ are non-adjacent (55 more relators). This Coxeter group (12 generators, 78 relators) is infinite, but by modding out by another strange `spider' relator

\begin{displaymath}
(a b_1 c_1 a b_2 c_2 a b_3 c_3)^{10} = 1
\end{displaymath}
the resulting quotient $N$ turns out to be a [[finite group]]. Here $a$ is the central vertex of valence $3$, $b_1, c_1$ are on an arm of length $4$ with $b_1$ adjacent to $a$ and $c_1 \neq a$ adjacent to $b_1$; similarly for $b_2, c_2$ on the other arm of length $4$, and for $b_3, c_3$ on the arm of length $3$. See \href{http://www.maths.qmul.ac.uk/~jnb/web/Pres/Mnst.html}{here} if this is not clear.

It turns out that $N$ has a [[center]] $C$ of order $2$, and the Monster $M$ is the quotient, i.e. the indicated term in the exact sequence

\begin{displaymath}
1 \to C \to N \to M \to 1.
\end{displaymath}
This implicitly describes the Monster in terms of 12 generators and 80 relators.

Such ``$Y$-group'' presentations (Coxeter group based on a similar $Y$-diagram, modulo a spider relation) are linked to a number of finite simple group constructions, the most famous of which is perhaps $Y_{555}$ which is a presentation of the ``Bimonster'' (the [[wreath product]] of the Monster with $\mathbb{Z}/2$). See \hyperlink{Iv}{Ivanov} for a general description of these. The presentation of the Monster given above was established in \hyperlink{Iv2}{Ivanov2}.

\hypertarget{ViaAutomorphisms}{}\subsubsection*{{Via automorphisms of a super vertex operator algebra}}\label{ViaAutomorphisms}

There is a [[super vertex operator algebra]], the [[Monster vertex operator algebra]], whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]].

(\hyperlink{FrenkelLepowskiMeurman89}{Frenkel-Lepowski-Meurman 89}, \hyperlink{GriessLam11}{Griess-Lam 11})

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

\begin{itemize}%
\item [[Moonshine]],


\item [[Monster vertex algebra]]


\item [[Mathieu group]], [[Mathieu moonshine]]



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

\begin{itemize}%
\item \href{http://mathoverflow.net/users/39521/adam-p-goucher}{Adam P. Goucher}, \emph{Presentation of the Monster Group}, (\href{http://mathoverflow.net/q/142216}{MO comment 2013-09-15})


\item Alexander Ivanov, \emph{Y-groups via transitive extension}, Journal of Algebra, Volume 218, Issue 2 (August 15, 1999), 412--435. (\href{http://www.sciencedirect.com/science/article/pii/S0021869399978821}{web})


\item A. A. Ivanov, \emph{Constructing the Monster via its Y-presentation}, in Combinatorics, Paul Erds is Eighty, Bolyai Society Mathematical Studies, Vol. 1 (1993), 253-270.


\item [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, \emph{Vertex operator algebras and the monster}, Pure and Applied Mathematics \textbf{134}, Academic Press, New York 1998. liv+508 pp. \href{http://www.ams.org/mathscinet-getitem?mr=996026}{MR0996026}


\item [[Robert Griess]] Jr., Ching Hung Lam, \emph{A new existence proof of the Monster by VOA theory} (\href{https://arxiv.org/abs/1103.1414}{arXiv:1103.1414})


\item [[Andre Henriques]], \emph{\href{http://mathoverflow.net/questions/69222/h4-of-the-monster}{$H^4$ of the monster}}



\end{itemize}
[[!redirects Monster]]

[[!redirects monster group]]

[[!redirects Monster simple group]]



\end{document}