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$^*$) There is a model of the Scott sentence of $M$ of size $\kappa^+$. (II$^*$) 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$^*$) For every $\beta<\kappa^+$, there exist $L_{\infty,\kappa}^{fin}$- 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$^*$), which does not appear in the countable case, can not be removed when $\kappa$ is uncountable (cf. theorem 4.5). Condition (II$^*$) 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.