javier wrote: > On Jul 31, 4:47 pm, David Joyner <[email protected]> wrote: >> Maybe I don't understand your question. It seems you are claiming >> thatif G is a permutation group and H is a normal subgroup then >> the quotient G/H embeds into G. Are you sure that is true? > > In general, no it isn't. We have the short exact sequence > 1 --> H --> G --> G/H -->1 > to embed G/H into G we would need it to split, which would mean that > the extension is trivial, or that G factorizes as a product of H and > G/H.
OK, my fault, too much wishful thinking here. kcrisman wrote: > > Look at > > sage: G.quotient_group?? > > It turns out that Sage asks GAP to create the image of the morphism G - >> G/H, as far as I can tell, and in so doing creates that image as a > separate (sub)permutation group. In particular, it using > RegularActionHomomorphism to do this, and at > http://www.gap-system.org/Manuals/doc/htm/ref/CHAP039.htm#SSEC007.2 it > says "returns an isomorphism from G onto the regular permutation > representation of G" and certainly in this case G/H (the relevant > group) has six elements! > > Though I agree that this could be confusing, the good part is that > this creates (an isomorphic) group without having to talk about which > element of the coset you pick each time. It would be misleading to > say that (1234) was an element of G/H (which I think is what David was > getting at). There are ways to get cosets in GAP, of course (maybe > wrapped in Sage?) but I don't know much about them. > > I hope this helps! > Yes, the cosets are, what I really want (although generators would be very nice, too, if they exist, e.g. if the quotient is normal). The application was the following: Suppose you have a linear equation with multi-indexed variables, like x_1,2 + x_2,3 + x_3,1 == 0, which also holds for all permutations of {1, 2, 3}, and I want to enumerate all possible equations, but without duplicates. I hoped it was possible to first compute the permutations under whose operation the exact same equation result, then take the subgroup H generated by those and use representants from the cosets of S_n/H to get all unique equations. Looks like it's not that simple, since H doesn't even have to be normal, in general. Thanks a lot , though. -- Robert Schwarz <[email protected]> Get my public key at http://rschwarz.net/key.asc --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
