> < ^ Date: Fri, 25 Jan 2002 20:44:04 -0500
> < ^ From: Igor Schein <igor@txc.com >
> < ^ Subject: Re: Subgroup question

Dear Gap Forum,

On Fri, Jan 25, 2002 at 02:14:20PM -0700, Alexander Hulpke wrote:
> Dear Gap Forum,
> Igor Schein wrote:
> > Thanks a lot, that's exactly what I needed.  Now I can answer my original
> > question:
> > 
> > gap> IsomorphicSubgroups(SmallGroup(32,29),SmallGroup(8,4));
> > [ [ f1, f2 ] -> [ f1, f2 ], [ f1, f2 ] -> [ f1*f5, f1*f2 ] ]
> > 
> > So SmallGroup(32,29) indeed contains a quaterunion subgroup.
> As the person who wrote this function, let me just add that
> `IsomorphicSubgroups' tries to find possible images of a generating set. If
> the group you want to embed has many generators (for example if it is
> elementary abelian) this is likely to become slow and a method more along
> thelines of Joachim Neub"users first suggestion will be better.
> > I don't understand this though:
> > 
> > \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
> > gap> IsomorphicSubgroups(SmallGroup(32,29),SmallGroup(4,1));
> > List Element: <list>[1] must have an assigned value at
> > return i[1];
> > Why error?
> I was not able to reproduce the problem in 4r2fix8, but if I remember
> correctly we had such a problem (with cyclic groups?) a while ago which
> got corrected in a bugfix. Could you check whether you have up to fix 8
> installed?  (If yes, and the problem still arises, please esend us a mail at
> gap-trouble@dcs.st-and.ac.uk and we will investigate.)
> Best regards,
>   Alexander Hulpke

True, I didn't have fix8 installed, but now I do, and everything is
fine. Now here's something that's either a embarassing
misunderstanding by me or a serious problem. I examined
SmallGroup(16,1) through SmallGroup(16,14) using IsomorphicSubgroups,
and determined that that each group may contain TransitiveGroup(8,3)
or TransitiveGroup(8,5), among others, but never both. However, I can
present a degree-16 polynomial, whose Galois group is of size 16, and
which contains subfields with both Galois group TransitiveGroup(8,3)
and TransitiveGroup(8,5). If some1 can explain me how this is NOT a
contradiction, I'd really appreciate it, because everything I've been
doing in the past year might go down the drain :-)

In case some1 wants to take a look at the polynomial, here it is:


I used both PARI/GP and Kant to verify my statement about the
subfields of this polynomial.



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