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.