beta G\] $(G,\beta G, e)$ is the unique up to isomorphism universal $G$-ambit.

Using this fact, one gets an “action” of $\beta G$ on any $G$-flow $(G,X)$, namely for $x \in X$ there is a unique flow homomorphism $h_x\colon (G,\beta G, e) \to (G,X,x)$, and for $p \in \beta G$ we define $px=h_x(p)$. More explicitly, this action is given by $px=\lim g_ix$ for any net $(g_i)$ of elements of $G$ converging to $p$ in $\beta G$. In particular, $\beta G$ acts on itself, and denoting this action by $*$, one has $(p*q)x=p(qx)$ for all $p,q \in \beta G$ and $x \in X$. In particular, $*$ is a semigroup operation on $\beta G$ which is continuous on the left and whose restriction to $G \times G$ is the original group operation on $G$. One easily checks that $(\beta G,*) \cong E(\beta G)$ (by sending $p \in \beta G$ to the function $(x \mapsto px) \in E(\beta G)$). In particular, Fact \[Ellis theorem\] applies to $(\beta G, *)$ in place of $E(\beta G)$.\
In this paper, $M$ denotes a model of a first order theory in a language $\mathcal L$, and ${{\mathfrak C}}\succ M$ is a monster model; $G$ will be a 