On a problem about tensor products of subfactors

In this paper we make progress on the tensor product conjecture about minimal intermediate subfactors in tensor products of subfactors which are not of tensor product form. This conjecture is motivated by a subfactor generalization of Wall's conjecture from the theory of finite groups. We reduce the tensor product conjecture to a conjecture about a class of what we call very simple Kac algebras. Such a reduction gives a proof of tensor product conjecture for group-subgroup subfactors. At present the only known such Kac algebras come from simple groups with possible twist, and we verify our conjecture in such cases.