The existence of \(J_ 3\) and its embeddings in \(E_ 6\) (Q915852)

The author determines the embeddings of the third sporadic group \(J_ 3\) of Janko in simple Chevalley groups of type \(E_ 6\) over finite and algebraically closed fields. As a corollary he obtains a short elegant existence proof of \(J_ 3\). This is of interest as \(J_ 3\) is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs (see the papers of \textit{G. Higman} and \textit{J. McKay} [Bull. Lond. Math. Soc. 1, 89-94 (1969; Zbl 0175.30103)] and \textit{R. Weiss} [Math. Z. 179, 91-95 (1982; Zbl 0481.05034)]) were highly computational. P. Kleidman and R. Wilson were the first to prove that \(J_ 3\) embeds in \(E_ 6(4)\). Their proof was machine aided.