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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… 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?" – Bill Cook Nov 23 '11 at 12:43
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. – Jyrki Lahtonen Nov 23 '11 at 12:47
I think a question should be judged by its own merit. Whether it is homework or not is of secondary importance at most. – Makoto Kato Aug 9 '12 at 19:08
@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. – anon Aug 9 '12 at 19:38
@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. – anon Aug 9 '12 at 20:14

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 $\,\mathcal Orb(s):=\{gs\;:\;g\in G\}\,$, many $\,G-\,$orbits are there?

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