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The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the following\par Theorem. Let $(M_i, \alpha_i)$, $i= 1,2$, be a geometric structure such that its group of automorphisms $G(M_i, \alpha_i)$ satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3', 4, 5, 6 and 7, and $M_i$ is compact, or axioms 1, 2, 3', 4, 5, 6, 7, 8 and 9. Then if there is a group isomorphism $\Phi: G(M_1, \alpha_1) \to G(M_2, \alpha_2)$ then there is a unique $C^\infty$-diffeomorphism $\varphi: M_1\to M_2$ preserving $\alpha_i$ and such that $\Phi(f) =\varphi f\varphi^{-1}$ for each $f\in G(M_1, \alpha_1)$. \par The axioms referred to in the theorem concern a finite open cover of $\text {supp} (f)$, $\text {Fix} (f)$, leaves of a generalization folia!
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