# About rotations of sets of vertices of a regular $p$-gon.

This is something I've been thinking about lately, and I don't seem to understand the problem well enough. There is a motivation to this problem, but I don't think giving it would be productive since the motivation is probably much more exotic and quite possibly less interesting than the problem.

Let $p$ be a prime and consider a regular $p$-gon $P$. Let $V$ be the set of vertices of $P$. I want to take any nonempty subset $T_0\subseteq V$ and define $T_i\subseteq V$ to be the set $T_0$ rotated by $\frac{2i\pi}{p}$ for $0\leq i\leq p-1.$ Thanks to $p$ being prime, $T_i$ are pairwise different, and look like this for $p=5$ and a specific $T_0$ (the rotations are clockwise here, which doesn't make any difference):

Now I want to take unions of these sets. That is, I want to consider the family $$\mathcal F(T_0)=\left\{\bigcup_{i\in I} T_i\,|\,\varnothing\neq I\subseteq\{0,1,\ldots,p-1\}\right\}.$$

I would like to understand what this family looks like for a given set $T_0$. Clearly if $T_0$ has only one element, then $\mathcal F(T_0)$ contains all nonempty subsets of $V$. But this is just about all I can see here.

So my first question is whether we can describe $\mathcal F(T_0)$ for a given $T_0$ in any understandable way.

More specifically, can we give the cardinality of $\mathcal F(T_0)$ for a given $T_0$? Does it depend on anything other than the cardinality of $T_0$ and the prime $p$?

And lastly, does this ring any bells? As I said, I came across this in a rather exotic setting and I can see no strong associations with anything standard I know. Of course this can all be said in terms of $\Bbb Z/p\Bbb Z$ and that's where I found this originally, but this doesn't change the fact that I have no idea what to do with it.

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I think this is a very interesting problem. Alternatively stated: Let $C_p$ act its power set by left translation and consider the closure of the power set of $\mathcal{O}_{T}$ under unions. – Alexander Gruber Dec 12 '12 at 9:25