On Aug 18, 8:24 pm, David Joyner <wdjoy...@gmail.com> wrote:
> On Thu, Aug 18, 2011 at 3:12 PM, John Cremona <john.crem...@gmail.com> wrote:
> > I wanted to work in the group PGL(2,q), and got off to a good start:
>
> > sage: G = PGL(2,13)
> > sage: G.order().factor()
> > 2^3 * 3 * 7 * 13
> > sage: G.order() == 13*(13^2-1)
> > True
>
> > but I could not create elements of G, which seemed to think they were
> > permutations!
>
> I believe this is the way GAP does things.
>
And not just Gap, Magma too:
> PGL(2,13);
Permutation group acting on a set of cardinality 14
Order = 2184 = 2^3 * 3 * 7 * 13
(3, 9, 10, 12, 6, 13, 4, 11, 5, 7, 8, 14)
(1, 14, 2)(3, 8, 13)(4, 10, 11)(5, 6, 12)
How strange.
>
>
> > sage: G.identity()
> > ()
> > sage: G.an_element()
> > (3,14,13,12,11,10,9,8,7,6,5,4)
> > sage: type(G.an_element())
> > <type 'sage.groups.perm_gps.permgroup_element.PermutationGroupElement'>
>
> > Now I am not a group theorist, but this just seems bizarre! I am in
> > fact quite interested in the action of this G on P^1(GF(13)), but
> > also expected to be able to work with its elements as matrices (mod
> > scalars).
>
> > Am I doing something wrong?
>
> I don't think so, but perhaps your problem can be translated into
> one regarding GL(2,13)?
>
Well that seems unnatural to me. I do want PGL(2) to act on the
projective line, though I would rather have its elements labelled more
sensibly than by the integers from 1 to 14. But I also want to see it
as a matrix group (mod scalars).
I was actually wanting to write down the subgroups (isomorphic to) A4
and A5 in PGL(2,q) (for suitable q). They do not lift up to GL(2,q)
so it's a bit inconvenient to work there. Never mind.
John
>
> > John
>
> > [Sage 4.7 on ubuntu, built from source]
>
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