Dieudonn\'e modules and $p$-divisible groups associated with Morava $K$-theory of Eilenberg-Mac Lane spaces

We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonn\'e correspondence between bicommutative Hopf algebras and Dieudonn\'e modules. We extend P. Goerss's results on the bilinear products of such Hopf algebras and corresponding Dieudonn\'e modules.