Symmetric Presentations II: The Janko Group J 1

Abstract

In our paper [6] we exploited parallel behaviour of the Mathieu groups M12 and M24 which we described fully in [4, 5], to obtain presentations of those groups given implicitly in terms of certain highly symmetric generating sets. We made precise the concepts of symmetric generating set, symmetric presentation and primitive presentation, and indicated how these ideas can be generalized to other groups. In the present work, which is a continuation of [6], we turn the process around and seek groups which possess generating sets with certain specified symmetries. We shall find that the most straightforward ways of generalizing [6] immediately give rise to the two smallest Janko groups J1 and J2 Moreover the permutation actions of minimal degree of these two groups, on 266 and 100 letters respectively, emerge naturally and we are readily able to write down our symmetric generators in these representations. All assertions made can be easily checked on a computer, and we are again heavily indebted to the CAYLEY package. That said the final developments of J1 and J2 given here, and in the sequel to this paper, are entirely by hand and are complete in all detail.