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A well-known result, Cayley's theorem, says that any group is isomorphic to $S_n$ for some $n$. Given a (finite) group $G$, is there a standard name for the smallest such $n$?

This seems like a very basic concept which has surely been studied and named, but my searches have found nothing so far.

Is it easy to calculate $n$? An obvious lower bound is the Kempner number A002034$(|G|)$.

Note: there is a related question but it doesn't address my issue.

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Charles
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    I'd like to suggest that you change the phrasing to "isomorphic to a subgroup of $S_n$". – Zev Chonoles Aug 02 '13 at 05:42
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    Equivalently, you're asking for what the size of the smallest set that $G$ acts faithfully on. I've had this thought before, but don't remember coming to a conclusion--good question! – Alex Youcis Aug 02 '13 at 05:45
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    It appears this has been asked on overflow: http://mathoverflow.net/questions/48928/smallest-n-for-which-g-embeds-in-s-n – Alex Youcis Aug 02 '13 at 05:46
  • The term seems to be "minimal representation of a finite group", one link http://www.jstor.org/discover/10.2307/2373739?uid=3739832&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102519829547 [But it isn't free unless you're on JSTOR] – coffeemath Aug 02 '13 at 05:47
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    as far as all finite groups are concerned, Cayleys theorem gives best estimate for $n$, since $Q_8$ can not be embedded in $S_7$. – Beginner Aug 02 '13 at 07:26
  • There is considerable literature on the subject, look up minimal degree for a permutation representation: http://goo.gl/TQAcZL – Andreas Caranti Aug 02 '13 at 10:35
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    http://math.stackexchange.com/questions/191446/efficient-version-of-cayleys-theorem-in-group-theory is another duplicate – Jack Schmidt Aug 02 '13 at 15:42

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