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Let G be a compact, connected Lie group and H a closed subgroup of G. It is shown that if GjH is highly connected relative to Rk(G) -Rk(H), GjH splits as a product of homogeneous spaces of simple Lie groups. This is used to show that the only transitive, effective actions on a large class of products of spheres are products of the known actions on the individual spheres.doi:10.1090/s0002-9939-1973-0321125-9 fatcat:wchajqvz3zgbtm3yienoi3phd4