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Given a finite group $G$, how efficient can one make an algorithm to find the size of the smallest set $S$ such that $G$ is isomorphic to a group of permutations of the members of $S$? And does the answer change if one requires the output to specify not only the cardinality of $S$ but the particular action of $G$ on $S$? Might this be an NP-hard problem? Or is it a trivial thing whose solution is known to everyone on earth except me? Or somewhere in between?

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What's trivial are the bounds that $\#G$ divides $\#S!$ and $\#S\le \#G$ or that the answer for cyclic $G$ of order $\prod p^{k_p}$ is $\sum p^{k_p}$, bu tbeyond that things can get messy ... – Hagen von Eitzen Oct 11 '13 at 20:26
I think this is a difficult problem, and that there is no efficient algorithm known. It might be easier to find an action that was close to being optimal. – Derek Holt Oct 11 '13 at 21:04
Slightly less trivial is that $|S|<|G|$ except when $G$ is prime cyclic or generalized quaternion. I do remember a question about this very thing being asked on math.SE a while ago, and @JackSchmidt giving a nice summary of what's known. But I cannot find it now. – user641 Oct 11 '13 at 23:28
Here are Jack Schmidt's answers, apparently mostly on group theory: – Michael Hardy Oct 11 '13 at 23:50
Also see this :… – Prahlad Vaidyanathan Oct 12 '13 at 4:04
up vote 4 down vote accepted

See this MO answer for links to several important papers.

The main citation is: Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866.

Edit (10/30/13): Check the comment below for an entire book on this subject.

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I now find what appears to be a whole book addressing this question: Neil Saunders, Minimal Faithful Permutation Representations of Finite Groups, University of Sydney, 2011… PS: OK, I now see that it's a Ph.D. thesis. It turned up in a Google Books search. – Michael Hardy Oct 30 '13 at 23:33

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