19.64 (2018)

Let $G$ be a group, and $(g_1, \dots, g_n)$ a tuple of its elements. The type of this tuple in $G$, denoted $Tp^G(g_1, \dots, g_n)$, is the set of all first order formulas in free variables $x_1, \dots, x_n$ in the standard group theory language which are true on $(g_1, \dots, g_n)$ in $G$. Two groups $G$ and $H$ are called isotypic if for every tuple of elements $\bar{h} = (h_1, \dots, h_n)$ in $H$ there is a tuple $\bar{g} = (g_1, \dots, g_n)$ in $G$, such that $Tp(\bar{h}) = Tp(\bar{g})$ and vice versa, for every tuple $\bar{g}$ in $G$ there is a tuple $\bar{h}$ in $H$ such that $Tp(\bar{h}) = Tp(\bar{g})$. Is it true that every two isotypic finitely generated groups are isomorphic?