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We construct a non-abelian extension Γ of  S  1  by  Z /3 ×  Z /3, and prove that Γ acts freely and smoothly on  S  5  ×  S  5 . This gives new actions on  S  5  ×  S  5  for an infinite family  of finite 3-groups. We also show that any finite odd-order subgroup of the exceptional Lie group  G  2  admits a free smooth action on  S  11  ×  S  11 . This gives new actions on  S  11  ×  S  11  for an infinite family ℰ of finite groups. We explain the significance of these families , ℰ for the general existence problem, and correct some mistakes in the literature.

EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES