MathieuGroups 2009-01-12 03:29:56 2009-01-12 03:45:25 Definition $M_{24}$ $M_{23}$ $M_{22}$ $M_{12}$ $M_{11}$ % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\gc}{\mathcal{G}_{24}} \PMlinkescapeword{size} \PMlinkescapephrase{automorphism group} \PMlinkescapephrase{automorphism groups} The Mathieu groups are a \PMlinkescapetext{series} of 5 sporadic simple groups discovered by the French mathematician \'Emile L\'eonard Mathieu. They are usually denoted by $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$. They are defined as automorphism groups of certain Steiner systems, and the subscript denotes the size of the underlying set of the Steiner system. If $\Omega$ is a set of $n$ elements, then a $(t,k,n)$-Steiner system on $\Omega$ is a set $S$ of subsets of $\Omega$, each of size $k$, such that every subset of $\Omega$ of size $t$ is contained in a unique element of $S$. The automorphism group of the Steiner system is defined as the permutations of $\Omega$ which map $S$ to itself. There exists a (5,8,24)-Steiner system, and it is unique up to permutation of the elements of $\Omega$. It can be constructed as the set of octads of the extended binary Golay Code $\gc$. We denote it by $S(5,8,24)$ below. There exists a (5,6,12)-Steiner system, and it is unique up to permutation of the elements. It can be constructed as follows. Take $\Omega'$ to be a dodecad (element of weight 12) of $\gc$. Then the subsets of size 6 in $\Omega'$ which are contained in an octad of $\gc$ form a (5,6,12)-Steiner system. We denote it by $S(5,6,12)$ below. \section{Definition of the Mathieu groups} The group $M_{24}$ is the automorphism group of $S(5,8,24)$. It has order $244\,823\,040 = 2^{10} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11 \cdot 23$. The group $M_{23}$ is the subgroup of $M_{24}$ fixing a given \PMlinkescapetext{point} of $\Omega$. It is the automorphism group of a (4,7,23)-Steiner system). It has order $|M_{24}|/24 = 10\,200\,960 = 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 23$. The group $M_{22}$ is the subgroup of $M_{24}$ fixing two given \PMlinkescapetext{points} of $\Omega$. It is a subgroup of \PMlinkname{index}{Coset} 2 in the automorphism group of a (3,6,22)-Steiner system. It has order $|M_{23}|/23 = 443\,250 = 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11$. The group $M_{12}$ is the automorphism group of $S(5,6,12)$. It has order $95\,040 = 2^6 \cdot 3^3 \cdot 5 \cdot 11$. The group $M_{11}$ is the subgroup of $M_{12}$ fixing a \PMlinkescapetext{point} of $\Omega'$. It is the automorphism group of a (4,5,11)-Steiner system. It has order $|M_{12}/12| = 7\,920 = 2^4 \cdot 3^2 \cdot 5 \cdot 11$. For further \PMlinkescapetext{information} on the Mathieu groups, consult the \PMlinkescapetext{references} (\cite{splag}, Chapters 10 and 11) and (\cite{sporadic}, Chapters 5-7). \emph{Note.} It is possible to continue the pattern above and define groups $M_{21}$, $M_{20}$, $M_{10}$, $M_9$, $M_8$. However, they are no longer sporadic simple groups. The group $M_{21}$ is a subgroup of \PMlinkescapetext{index} 3! = 6 in the automorphism group of a (2,5,21)-Steiner system, which are the points and lines of the projective plane over the field of 4 elements. In fact, $M_{21} \cong PSL(3,\mathbb{F}_4)$, a \PMlinkname{simple group of Lie type}{ProjectiveSpecialLinearGroup}. The group $M_{20}$ is a solvable group. The group $M_{10}$ is not simple; it contains, with \PMlinkescapetext{index} 2, the alternating group $A_6$ (sometimes denoted $M_{10}'$ in this context since it is the derived subgroup of $M_{10}$). The groups $M_9$ and $M_8$ are solvable. \begin{thebibliography}{9} \bibitem{splag} J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999. \bibitem{sporadic} Robert L. Griess, Jr. Twelve Sporadic Groups. Springer-Verlag, 1998. \end{thebibliography}