10

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?

Michael Hardy
  • 1
  • 1
    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
  • 4
    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
  • 1
    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. –  Oct 11 '13 at 23:28
  • Here are Jack Schmidt's answers, apparently mostly on group theory: http://math.stackexchange.com/users/583/jack-schmidt?tab=answers – Michael Hardy Oct 11 '13 at 23:50
  • Haha, I don't know that I would consider a Friday night spent combing through math.SE answers a worthwhile venture. :) –  Oct 12 '13 at 01:17
  • 1
    Also see this : http://math.stackexchange.com/questions/514650/on-group-theory-terminology – Prahlad Vaidyanathan Oct 12 '13 at 04:04

1 Answers1

4

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.

Community
  • 1
Benjamin Dickman
  • 16,038
  • 1
    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 http://books.google.com/books?id=JGoiuAAACAAJ&dq=%22Minimal+Faithful+Permutation+Representations+of+Finite+Groups%22&hl=en&sa=X&ei=rJZxUsajG6PhygGF5IGABg&ved=0CC4Q6AEwAA 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