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From: Derek Holt on 16 Feb 2010 16:15
On 16 Feb, 20:11, Kent Holing <K...(a)statoil.com> wrote:
> Can anybody give an example of equation of degree n with a Galois group of order 2n but not Dn?

x^6 - 19*x^4 + 104*x^2 - 169

(Galois group A_4)

Derek Holt.
From: Kent Holing on 16 Feb 2010 06:46
Thanks. Do you have such an example where the equation has both real and complex roots (and is irreducible)?
From: Jim Heckman on 17 Feb 2010 02:58

On 14-Feb-2010, Chip Eastham <hardmath(a)gmail.com>
wrote in message
<508a6f4a-04e9-434d-846f-faaf2a2322cb(a)k41g2000yqm.googlegroups.com>:

> Not all subgroups of order 12 in Sym(6) are dihedral,
> e.g. there are copies of Alt(4) in there, but all
> the transitive subgroups of order 12 are dihedral.

This will come as a surprise to the transitive subgroup ~= A_4 of
S_6 generated by, say, <(1,2)(3,4), (1,3,5)(2,4,6)>.

> I was trying to work in the point we saw earlier in
> the discussion that not all the the copies of Dih(6)
> in Sym(6) are transitive.

--
Jim Heckman
From: Derek Holt on 17 Feb 2010 04:27
On 16 Feb, 21:46, Kent Holing <K...(a)statoil.com> wrote:
> Thanks. Do you have such an example where the equation has both real and complex roots (and is irreducible)?

x^6 + x^4 - 2*x^2 - 1

has 2 real and 4 complex roots and Galois group A_4.

I found that one quickly by random search. I got the earlier one from
a database - I don't know why it has relatively large coefficients,
but perhaps the polynomials in the database were generated by some
algorithmic process.

Derek Holt
From: Chip Eastham on 17 Feb 2010 05:33
On Feb 17, 2:58 am, "Jim Heckman" <rot13(reply-to)@none.invalid>
wrote:
> On 14-Feb-2010, Chip Eastham <hardm...(a)gmail.com>
> wrote in message
> <508a6f4a-04e9-434d-846f-faaf2a232...(a)k41g2000yqm.googlegroups.com>:
>
> > Not all subgroups of order 12 in Sym(6) are dihedral,
> > e.g. there are copies of Alt(4) in there, but all
> > the transitive subgroups of order 12 are dihedral.
>
> This will come as a surprise to the transitive subgroup ~= A_4 of
> S_6 generated by, say, <(1,2)(3,4), (1,3,5)(2,4,6)>.

Thanks, Jim:

And apologies to the unsung copy of A_4
you describe, who deserves recognition
as a minimal example of a transitive
subgroup of order 2n in Sym(n) not
isomorphic to Dih(n)!

regards, chip


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