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I apologize if this seems elementary but I don't know how to deal with. Let $G$ be a group of order $n$ and let $ \emptyset \ne S \subseteq G$. Is it true that $S^n :=\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$ is a subgroup of $G$ !? For example if $S$ contains a single element then by Lagrange's theorem $S^n = \lbrace e \rbrace$.

Robert M
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    It seems really impolite to delete the question from MSE and post it here with no mention of this fact. Especially since there were several comments on MSE relevant to the question. – Tobias Kildetoft Aug 26 '13 at 10:10
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    I agree with Tobias Kildetoft. I would also like to know where this problem came from, and why you want to know the answer. –  Aug 26 '13 at 10:13
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    @DerekHolt I am new here and I don't know the regulations. Do you suggest that I move it back to MSE !? any why I want to know the answer ??! Isn't it clear ? Same story that all of us has experienced: I have a problem and I cann't solve it ! – Robert M Aug 26 '13 at 10:31
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    Whether or not to ask in on MO also is debatable (I am not sure if I agree with it, but I don't find it obviously wrong). But since the MSE question has valuable comments, it seems a shame to have those be wasted. So please undelete the MSE question and put a link between them (so people can see both and avoid duplicate effort). – Tobias Kildetoft Aug 26 '13 at 10:53
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    @DerekHolt Is this a court ?? –  Aug 26 '13 at 11:04
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    I fully agree with the two comments by Tobias Kildetoft and I find your reaction to Derek Holt also very rude. The fact that you want badly the answer to your question doesn't prevent you to be polite. Remember you are asking people to work for you. So please undelete your MSE question and be more respectful to the people that are trying to help you. – J.-E. Pin Aug 26 '13 at 11:09
  • @RobertM: When in doubt, flag for moderator attention and explain what you want to do. There are mechanisms set up to transfer questions from one site to another. – François G. Dorais Aug 26 '13 at 12:09
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    @Robert M: You haven't answered my question about the source of the problem. I could rephrase it as "why do you believe this question to be interesting?". It does seem to be difficult to answer, but lots of problems in mathematics are hard, and personally I am only willing to devote a lot of effort in trying to solve a problem if I have some motivation for doing so. –  Aug 26 '13 at 13:04
  • @FrançoisG.Dorais Thanks. I have undeleted the question. – Robert M Aug 26 '13 at 13:07
  • @DerekHolt I heard it from a friend. And if think there is not enough motivation then its okay. – Robert M Aug 26 '13 at 13:11
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    Although this does appear to be a challenging problem, I do wonder whether it it has somehow been incorrectly stated. If you replace the set $S^n$ by the set of all products of elements of $S$ of length at most $n$ (rather than exactly $n$ - in fact products of length at most $n-1$ would work too), then it becomes a sensible exercise. – Derek Holt Aug 26 '13 at 13:31

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