# Meaning of Strong Primitivity

As J.D.Dixon noted in his great book; Permutations Group, we can speak about Strong Primitivity of a group acting on a set $\Omega$ by means of orbital graphs. The way he paved employes digraphs prove if such a group is strong primitive or not. I referred to another reference came from P.J.Cameron (Toplogy in permutation groups) to find additional information. He cited that Wielandt observed that primitivity is not strong enough. Cameron continues by illustrating an example:

Let $G$ is a primitive group on a finite set $\Omega$ and let $\Delta$ be a non-empty proper subset of $\Omega$. then the following seperation property holds: for any distinct $\alpha$, $\beta$ $\in$ $\Omega$, there exists $g\in G$ with $\alpha^g$ $\in \Delta$ and $\beta^g$ $\notin \Delta$. This fails for infinite primitive groups.

and that is why Wielandt proposed the property of Strong Primitivity. Honestly, I didn't get much of what Cameron noted and need someone to elaborate. Why such a condition's named separate property? Thanks.

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## 1 Answer

The smallest $G$-invariant topology in $\Omega$ containing $\Delta$ is the topology generated by the translated sets $\Delta^g$ for all $g\in G$. Its basis consists of finite intersections of such sets. A point $\alpha$ in $\Omega$ has a neighborhood not containing a point $\beta$ if and only if $\alpha\in\Delta^g$ but $\beta\not\in\Delta^g$ for some $g\in G$. The existence of such a neighborhood for each pair $(\alpha,\beta)$ of distinct points of $\Omega$ is the separation property T1 of topology. The weaker property T0 means that this only holds for either $(\alpha,\beta)$ or $(\beta,\alpha)$ but not necessarily for both pairs.

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