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I have some difficulties proving next statement :

If $A$ is a block for a group $G$ and $a \in A$, show that $A$ is a union of orbits for $G_a$ (where $G_a$ is a stabilizer of a in G ).

I would be very thankful for some advice how to start.


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I'm unfamiliar with the term "block". Could you specify what the definition is? Also, could make more explicit the fact that the group $G$ is acting on the set $A$? – Zev Chonoles Jun 6 '11 at 19:56
@Zev: If $G$ acts on a set $S$, then a subset $A$ of $S$ is a block if for every $g\in G$ either $gA=A$, or $gA\cap A = \emptyset$. – Arturo Magidin Jun 6 '11 at 20:01
@Arturo: Ah, thanks. – Zev Chonoles Jun 6 '11 at 20:03

Hint. If $g\in G_a$, then $gA=A$ (since $ga = a\in gA\cap A$). Therefore, $G_aA = A$.

Now let $b\in A$. The orbit of $b$ for $G_a$ is $G_ab$. Is it contained in $A$? Why?

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Thank you so much. I got it. – aldo Jun 6 '11 at 20:19

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