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A group $G$ acts transitively on a non empty $G$-set $S$ if, for all $s_1,s_2 \in S$, there exists an element $g \in G$ such that $gs_1=s_2$. Characterize transitive $G$-set actions in terms of orbits. Prove your answer.

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    please read http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question It generally considered rude to ask for help by demanding "Prove your answer" Stackexchange is not a textbook but people kindly offering help. Please edit your question. Try rephrasing such as "I've tried so-and-so. Could you please help me with a proof?"2011-11-23
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    While I agree that it is 'bad form' to copy/paste an exercise here, I will also give a hint (this is very clearly homework): *it is about the number of orbits.*2011-11-23
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    I think a question should be judged by its own merit. Whether it is homework or not is of secondary importance at most.2012-08-09
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    @MakotoKato Your comment is tangential to this question. What upsets people is that textbook exercises are phrased in tone as commands directed at the reader, and so when askers copy/paste them without adding any of their own thoughts or attempts or context or anything else whatsoever, or even indicating it's a direct textual rip by using blockquotes, it means the wording is such that the OP is *commanding* his or her readers to do the exercise *for* them.2012-08-09
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    @anon While I understand your sentiment, I'm more interested in a mathematical question per se than the OP's intention whatever it is. For example, imagine that someone asked a question that $x^n + y^n = z^n$ has no integer solutions if $n \geq 3$ while people had never heard about FLT. Would it matter whether he wrote it like "prove this" or whether it was homework?2012-08-09
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    @MakotoKato I know what you're interested in and I was telling you how you misperceived the issue in the comments, not telling you of my personal sentiment (though it does align, if you wish to know). If you want a serious answer to the hypothetical: a homework or "prove this" question asking for a proof of FLT would be slammed for being a deliberate prank, as it is vanishingly unlikely it could be anything else. At any rate, you are using the comment thread on an old question to soapbox (speak at others, instead of truly with them) about tangential issues; I've overspent my attention here.2012-08-09
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    @anon Perhaps you misunderstand my comment. I said "while people had never heard about FLT". It's hypothetical because we all know it's FLT and it's a very important question. My point is that a question should be valued by its own merit. Regards,2012-08-10

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By the definition you gave, it seems like if we choose and fix an element $\,s\in S\,$, then $\,\forall\,s_1\in S\,\exists\,g\in G\,\,s.t.\,\,gs=s_1\,$ , and since the orbit of $\,s\,$ is defined to be $\,\mathcal Orb(s):=\{gs\;:\;g\in G\}\,$, then..how many $\,G-\,$orbits are there?