One of the most important products in permutation group theory is the wreath product, acting in its product action. The reason for this is that, unlike other products, it preserves a fundamental property called primitivity. Groups of wreath product type form an entire class in the classification of the finite primitive permutation groups (O'Nan–Scott Theorem).

I am going to talk about a new product which is fundamentally different to the wreath product in product action. Nevertheless, it preserves primitivity under astonishingly similar conditions. Moreover, under natural conditions on groups M and N, the product of M and N is simple. This fact means that the product can be used to easily solve a well-known open problem in topological group theory.