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I think that the following question is conceivably research-level, but it arose just as a curiosity, not part of any serious research. Is it reasonable to ask it on MO? (I include the specific question in case it affects your answer, but I am not intending to ask the question in this inappropriate forum.)

Suppose that $V_1$ and $V_2$ are vector spaces (over $\mathbb C$, say) with the same underlying set $X$. Suppose that the set of permutations of $X$ that are linear automorphisms of $V_1$ is the same as the analogous set for $V_2$. Then are $V_1$ and $V_2$ isomorphic?

EDIT: Stefan Kohl points out that it doesn't matter how the question arose, so probably I should clarify that I was asking because I don't know if a random curiosity is research-level—as opposed to something that arose in my research, which more or less by definition is (hopefully!).

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  • $\begingroup$ You might search MathOverflow for questions involving vector spaces and the axiom of choice. You may find something to help answer the proposed question. With choice, I think the answer is yes because the subset of Perm X "has enough torsion", but I do not know a convincing proof. $\endgroup$ – The Masked Avenger Nov 6 '14 at 23:53
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    $\begingroup$ My initial reaction is: great question; please ask it! I'd love to see an answer. $\endgroup$ – Joel David Hamkins Nov 7 '14 at 2:43
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    $\begingroup$ @JoelDavidHamkins, thanks for the encouragement! It is at mathoverflow.net/q/186439/2383. $\endgroup$ – LSpice Nov 7 '14 at 2:52
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    $\begingroup$ I think people should in general be less worried about asking questions that aren't at high enough of a level. If you're considering that issue, you're not the sort of person that rule is meant to exclude (it seems). Moreover, it's not the end of a world to have a question closed or transferred. If it happened because it's too easy, that's kind of an answer in itself. So just ask! $\endgroup$ – Jonathan Beardsley Nov 10 '14 at 4:25
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    $\begingroup$ Further to @JonBeardsley's comment: I always am troubled by using "level" as an indicator, since it is possible to ask very fruitful or thoughtful questions that only require definitions one would see as an undergraduate, and possible to ask very inane questions about sheaves, von Neumann algebras, cohomology, whatever $\endgroup$ – Yemon Choi Nov 14 '14 at 13:21
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    $\begingroup$ @YemonChoi, then perhaps you will enjoy my upcoming question about the cohomology of sheaves of von Neumann algebras? :-) $\endgroup$ – LSpice Nov 14 '14 at 13:50
  • $\begingroup$ @LSpice It depends if it's inane, or thoughtful :) $\endgroup$ – Yemon Choi Nov 14 '14 at 14:07
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    $\begingroup$ Some highly upvoted questions asked by some of our most illustrious users are more or less asked out of simple curiosity. You are in good company here. $\endgroup$ – Todd Trimble Nov 14 '14 at 14:30
  • $\begingroup$ To illustrate my previous remark: my view is that mathoverflow.net/questions/187303/… is a much better question than mathoverflow.net/questions/187246/… $\endgroup$ – Yemon Choi Nov 16 '14 at 12:41
  • $\begingroup$ in interests of fairness, vote this comment up if you disagree with the view in my previous comment $\endgroup$ – Yemon Choi Nov 16 '14 at 12:42
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Many people like to know where a problem came from. But don't be afraid to reply that you just had some idea and found it interesting. We mathematicians can identify with that. Of course we might not agree with you on how interesting the problem is, and some of us (not me) will be eager to tell you so.

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It matters what your question is and how you pose it, but not how you got to it. -- It doesn't make your question any worse or any better if you got to it in this or in that way.

Questions turning up in one's research are sometimes even not the best-suited ones for MathOverflow. Answering such questions frequently tends to be more work than an answerer on this site is likely to do, and answers are often longer than one or two pages of text, and thus rather too long for the format of this site.

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  • $\begingroup$ Good point—so maybe I should clarify why I asked. I'll update my question accordingly. $\endgroup$ – LSpice Nov 7 '14 at 0:46
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    $\begingroup$ I would say that while this is true in this case, it does not always hold. There are many questions that need the context of a larger research problem to be interesting (rather than way too specific for example). $\endgroup$ – Tobias Kildetoft Nov 7 '14 at 8:09

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