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There are many previous studies on the Hopf algebra $K(n)_*(K(n))$, thestable cooperations of $n$th Morava $K$-theory at an odd prime. Whereas themain part of $K(n)_*(K(n))$ corepresents the group-valued functor consisting ofstrict automorphisms of the Honda formal group law of height $n$, relationsbetween the whole structure of $K(n)_*(K(n))$ including the exterior part andformal group laws have not been investigated well. Firstly, we constitute afunctor $C(-)$ which is given by the fiber product of two natural homomorphismbetween subgroups of automorphisms of formal group laws, and the Hopf algebra$C_*$ corepresenting $C(-)$. Next, we construct a Hopf algebra homomorphism$\kappa^*:C_*\to K(n)_*(K(n))$ naturally. To relate $C_*$ to $K(n)_*(K(n))$, weuse stable comodule algebras which are introduced by Boardman. From the algebrastructure of $K(n)_*(K(n))$ which is given by W\"urgler and Yagita, we see that$\kappa^*$ is an isomorphism. Since we formulate $C_*$ by using formal grouplaws, the isomorphism $\kappa^*$ clarifies relationship between the Hopfalgebra structure of $K(n)_*(K(n))$ including the exterior algebra part andformal group laws.

The stable cooperations of Morava $K$-Theory and the fiber product of automorphism groups of formal group laws