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Orbits of an action of $G$ on a set $X$

Let $G$ be a finite group of order $p^\alpha q^\beta$ where $p, q$ are distinct primes and $\alpha,\beta>0$. Let $X$ be a set of size $p^\alpha$ and $G$ acts on $X$ via the action map $G\times X\to X,\, (g,x)\mapsto g.x$.

Prove that there exists a subset $Y$ of $X$ such that $|\mathcal{O}_G(Y)|=q^\beta$.

$\textbf{My attempt:}$
Since the group $G$ is not cyclic, it is not necessarily true that a non-trivial element always has exactly a single orbit. Hence, $|\mathcal{O}_G(Y)|>1$ for any $Y\subseteq X$. My idea is to prove that $|\mathcal{O}_G(Y)|$ is independent of $Y\subseteq X$ as long as $Y$ is chosen such that $|Y|\geq q^\beta.$
Clearly, $Y
eq \emptyset$. Now, a $(g_1,g_2)\in G\times G$ maps $Y\times Y$ to

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A new mini-game in which you will have the idea to drive…

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This game actually works like the perfect game, but with this addition.

“Tactics Game Dragons real test v. Dragons.”

In this game, you will have the idea to make a dragon as big as you can.
This mini-game is very interesting because if you did nothing at all, the dragon will become slow, he will simply be a ship that goes back and forth in order to get to the end of the level.
However, if you push him hard, he will become very fast, so you will have the idea to keep him in high gear, but not too high because you have a limited amount of fuel.
At the same time, you will have to choose a very clear route because, at every level, you will have the idea to avoid the obstacles that will appear at the end of the level.

Some new features in the Superb game.
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