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[sage-devel] Re: possible bug in permutation/quotient group? kcrisman Fri Jul 31 09:30:59 2009



On Jul 31, 11:47 am, David Joyner <[EMAIL PROTECTED]> wrote:
> Maybe I don't understand your question. It seems you are claiming that
> if 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?
>
>
>
> On Fri, Jul 31, 2009 at 11:36 AM, Robert Schwarz<[EMAIL PROTECTED]> wrote:
>
> > Hi all, I was just playing around with permutations, when something
> > puzzled me:
>
> > sage: G = SymmetricGroup(4)
> > sage: H = G.normal_subgroups()[1]
> > sage: H
> > Permutation Group with generators [(1,3)(2,4), (1,4)(2,3)]
> > sage: G.quotient_group(H)
> > Permutation Group with generators [(1,2)(3,6)(4,5), (1,3,5)(2,4,6)

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!

- kcrisman

>
> > Where do the 5 and 6 suddenly come from? In my understanding the
> > elements of the quotient group G/H are classes of elements of G, which
> > operates on {1, 2, 3, 4}.
>
> > Also, there is a method of G called "quotient", which raises and
> > NotImplementedError, which is a little confusing, given an
> > implementation of the quotient group is actually available.
>
> > Running Sage 4.1 on Arch Linux 64 bit.
>
> > --
> > Robert Schwarz <[EMAIL PROTECTED]>
>
> > Get my public key athttp://rschwarz.net/key.asc
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