Computations of Orbits for the Lubin-Tate Ring

We take a direct approach to computing the orbits for the action of the automorphism group ${\mathbb{G}}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{$\mathbb{G}$_2} \cong $\mathbb{F}$_p$\mathrm{.}Theresultisnewfor$p=2$and$p=3$\mathrm{.}Forprimes$p\geq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.