Groups which are cogroups

Abstract

It is the purpose of this note to give an elementary proof that S, S1, S3 are the only spheres which can be made into topological groups. This was proved by Samelson in [3]. He showed that a compact Lie group which is a homotopy sphere must have rank 1, and then that a compact Lie group of rank 1 has dimension 1 or 3. We get the first part easily by showing that a compact Lie group which is an ii-cogroup (e.g., a suspension) must have rank 1. The second part closely follows Samelson's proof. For basic facts about u-groups and u-cogroups the reader is referred to §§5 and 6 of Chapter I of Spanier [S]. If G is an u-group which is also an u-cogroup, then the set [G, G]o of pointed homotopy classes is an abelian group with its operation given equally by the u-cogroup structure on the domain G or the u-group structure on the range G [S, page 44]. From this the following lemma is immediate.