Dear Gap Forum and Istvan:
From: I. T. Hernadvolgyi[SMTP:firstname.lastname@example.org]
Sent: Wednesday, August 06, 1997 8:40 PM
To: Multiple recipients of list
Subject: Size of Minimal Generating Sets
I am wondering whether some bound on the number of elements of
minimal generating sets is known. I seem to recall a theorem, I
believe due to Erdos, which would suggest that if G is a finite permutation
group then any minimal generating set for G has at most O(log |G|)
If G is a subgroup of Sn (the symmetric group on n letters) with
minimal generating set of size b then
2^b<=|G|<=n(n-1)...(n-b+1)<=n^b. (Dixon and Mortimer, Permutation
Groups, Springer-Verlag, page 76, section 3.3.
> This of course is not true if G has too many orbits, but is it true for
> (almost) transitive groups?
It's true for all finite permutation groups.
> The other question I have is, if it is known to be NP-complete to find the
> shortest word between two arbitrary elements in a permutation group for
> a given generating set.
I think this is not true, if I understand your question correctly. (It depends on what parameter you're using to measure the speed of the algorithm.) In general, for a graph with N vertices the shortest path problem has an O(N^3) - time algorithm. See Gibbons, Algorithmic Graph Theory, Cambridge Univ Press, section 1.4, page 32. The Cayley graph of G has |G| vertices, of course, so this algorithm is polynomial in |G| but exponential in b. - David Joyner > > > References are greatly appreciated. > >thanks, > >-- > > - Istvan > > Istvan T. Hernadvolgyi | WEB: http://www.csi.uottawa.ca/~istvan > email@example.com | home: ( 613 ) - 237 - 0168 > firstname.lastname@example.org | office: MCD 214, Tel: 562-5800 ext: 6782 > > Former Department of Computer Science > School of Information Technology and Engineering > University of Ottawa > Ottawa, Ontario K1N 6N5, Canada > >