Dear Martin Can I suggest as a canonical form for a conjugacy class of permutation group G \leq S_n
1. Fix once and for all an ordering on the permutations in S_n - lex ordering when storing the permutations as their image lists would do. 2. Find the minimal number of generators d of G. 3. Store a lex-minimal generating set amongst the different ones of size d? I’m not completely sure this would work, and it would be awkward to compute, but it would give a canonical representative of each conjugacy class of groups? Best wishes Colva > On 18 Dec 2017, at 12:10, Thomas Breuer <s...@math.rwth-aachen.de> wrote: > > Dear Martin, > > coming back to an initial question asked by Alexander, > your examples seem to indicate that *isomorphism as abstract groups* > is not the appropriate notion of equivalence. > > When groups arise as symmetries of finite sets such as the vertices > of graphs then it is more natural to consider *permutation isomorphism* > (that is, conjugacy in the symmetric group on the given points). > For example, a group of order two can act on four points > by swapping two pairs or by fixing two points and swapping the other > two points; these two possibilities should probably be distinguished > in such a context. > > With respect to permutation isomorphism, groups are considered as small > when they are permutation groups on a small set, regardless of their > group orders. > GAP's library of transitive groups provides a reasonable source of > small groups in this sense. > > All the best, > Thomas > > > On Sun, Dec 17, 2017 at 08:58:45AM +0100, Martin Rubey wrote: >> Dear Alexander Hulpke, Dear Forum, >> >> many many thanks for your comments! Let me try to clarify - I apologize >> for the lengthy text... >> >>> There is no fundamental obstacle, but you either will end up with just >>> referring to some of the libraries of groups, or end up with an >>> exceeding amount of work by hand to make things come out nicely: >>> >>> - What groups are you planning to classify? Abstract groups or >>> Permutation groups (i.e. group actions)? >> >> the idea is to have finite abstract groups in findstat. >> [...] > > > _______________________________________________ > Forum mailing list > Forum@gap-system.org > https://mail.gap-system.org/mailman/listinfo/forum ************************************* Colva Roney-Dougal Reader in Pure Mathematics Director of the Centre for Interdisciplinary Research in Computational Algebra Editor of Proceedings A of the Royal Society of Edinburgh Editor of Royal Society Open Science Director of Undergraduate Mathematics Admissions The University of St Andrews is a charity registered in Scotland : No SC013532 _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
