Janko groups JankoGroups 2003-10-07 06:11:39 2005-03-18 22:09:15 Definition sporadic groups % 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 \PMlinkescapeword{group} \PMlinkescapeword{algebra} \PMlinkescapeword{simple} \PMlinkescapeword{finite} \PMlinkescapeword{solvable} The Janko groups denoted by $J_1, J_2, J_3$, and $J_4$ are four of the 26 sporadic groups. They were discovered by \PMlinkescapetext{Z}. Janko in 1966 and published in the article "A new finite simple group with abelian Sylow $2$-subgroups and its characterization.'' (Journal of Algebra, \textbf{3}, 1966, 32: 147-186). Each of these groups have very intricate matrix representations as maps into large general linear groups. For example, the matrix $K$ corresponding to $J_4$ gives a representation of $J_4$ in $GL_{112}(2)$.