On dual centralizer relations for group extensions and solutions of binomial equations

In this paper, different types of centralizer relations for group extensions G/F are explored in some depth. The different types of centralizer relations considered cover dual relations between the intermediate groups of two group extensions defined by centralizer relations within a (usually) larger group A in which both group extensions are embedded. But other considered types also address stronger forms covering dual centralizer relations between isomorphic embeddings or automorphisms for one group extension and group elements in the other group extension with inner automorphisms inducing these isomorphisms or automorphisms. A notion of inner closure for a given group extension G/F is introduced with an existence result for it that guarantees the existence of the different types of centralizer relations for G/F in its inner closure. In the last part of the paper, the above notions are used to obtain some results on the number of solutions of binomial equations over groups.