On automorphisms groups of structures of countable cofinality

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in ${\omega }^{\omega }$. (III) There is an $L_{{\omega }_1,\omega }$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $\kappa$ of of cofinality $\omega$ (cf. theorem 4.2). The following are equivalent: (I${}^{\ast }$) There is a model of the Scott sentence of $M$ of size ${\kappa }^{+}$. (II${}^{\ast }$) For all $\alpha <\beta <{\kappa }^{+}$, there exist functions $j_{\beta ,\alpha }$ in ${\overline{Aut(M)}}^T\setminus Aut(M)$, such that for $\alpha <\beta <\gamma <{\kappa }^{+}$, \begin{equation}() j_{\gamma,\beta}\circ j_{\beta,\alpha}=j_{\gamma,\alpha},\end{equation} where ${\overline{Aut(M)}}^T$ is the closure of $Aut(M)$ under the product topology in ${\kappa }^{\kappa }$. (III${}^{\ast }$) For every $\beta <{\kappa }^{+}$, there exist $L_{\mathrm{\infty },\kappa }^{fin}$- elementary embeddings (cf. definition 2.5) $(j_{\alpha }{)}_{\alpha <\beta }$ from $M$ to itself such that $\alpha_1<\alpha_2\Rightarrow range(j_{\alpha_1})\subset range(j_{\alpha_2})$. Theorem 4.2 holds both for countable and uncountable $\kappa$. Condition () in (II${}^{\ast }$), which does not appear in the countable case, can not be removed when $\kappa$ is uncountable (cf. theorem 4.5). Condition (II${}^{\ast }$) imply the existence of at least ${\kappa }^{\omega }$ automorphisms of $M$ (cf. corollary 4.6). It is unknown to the author whether a purely topological proof of corollary 4.6 exists.