Bimodule structure of central simple algebras

For a maximal separable subfield K of a central simple algebra A, we provide a semiring isomorphism between K-K-sub-bimodules of A and H-H-sub-bisets of G=Gal(L/F), where F=Cent(A), L is the Galois closure of K/F, and H=Gal(L/K). This leads to a combinatorial interpretation of the growth of dimK⁡((KaK)i), for fixed a∈A, especially in terms of Kummer subspaces.