The existence of J₃ and its embeddings in E₆

Abstract

We determine the embeddings of the third sporadic group J₃ of Janko in simple Chevalley groups of type E₆ over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J₃. This is of interest as J₃ is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].

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