HS<E₇(5)

Abstract

In this paper we describe our discovery that the sporadic simple groups HS and M₂₂ are contained in the simple Chevalley group E₇(5).

The work of [9] produces a short list of the possibilities for a sporadic simple subgroup of an exceptional group of Lie type. Apart from possible embeddings of M₂₂ and HS in groups of type E₇ in characteristic 5, all of the embeddings of [9] are already known to occur. Thus our paper completes the classification of sporadic simple subgroups of exceptional groups of Lie type.

We give two proofs of the embedding HS<E₇(5). The first of these proofs is entirely computer free, while the second proof makes some use of machine calculations. As a step in our hand proof of HS<E₇(5) we establish the embedding M₂₂<E₇(5): of course, since M₂₂ is a subgroup of HS, this result also follows as a consequence of our computer proof of HS<E₇(5).

We were led to conjecture the inclusion HS<E₇(5) for the following reasons (similar arguments are presented in [9]). The double cover 2.HS has a faithful 56- dimensional character, whose values are compatible with the character values of groups of type E₇ acting on their natural 56-dimensional module. Now HS and 2.HS contain a subgroup 5²[ratio ]20, an elementary abelian group of order 25 extended by a cyclic group of order 20 acting faithfully on the 5². Since 20 is not the order of an element in the Weyl group W(E₇) = 2×S₆(2), it can be shown that 5²[ratio ]20 does not embed in groups of type E₇(K), where K is a field of characteristic prime to 5. Thus HS embeds in E₇(q) only if 5[mid ]q. On the other hand, all local subgroups of 2.HS embed in 2.E₇(5), where our conjecture HS<E₇(5).

Throughout, G denotes the double cover 2.E₇(5), G¯ denotes the simple group E₇(5), and V denotes the natural 56-dimensional module for G over GF(5). Most of our notation follows that of the atlas [4]. The two sections of our paper are completely independent.