[GAP Forum] Automorphism group of Aut(D2n)

[tkohl at math.bu.edu]

There are a couple of ways to do it:


--------------------------------------------
n:=3;; # or whatever value you wish
G:=DihedralGroup(2*n);
RG:=Action(G,Elements(G),OnRight);  # basically the right regular representation
AutG:=Action(AutomorphismGroup(G),Elements(G));
--------------------------------------------

Alternately, you can do this:
--------------------------------------------
n:=3;;
G:=DihedralGroup(2*n);
RG:=Action(G,Elements(G),OnRight);  # basically the right regular representation
HolG:=Normalizer(SymmetricGroup(2*n),RG);;
AutG:=AsGroup(Filtered(Elements(HolG),g->OnPoints(1,g)=1));;
--------------------------------------------


which is a bit more computationally intensive as n increases because
one is filtering the elements of a group (the holomorph).

However, it's nice in that it is a manifestation of one of the classic 
definitions of the automorphism group, namely as the set of those elements 
(permutations of G as a set) which normalize the right (or left) regular 
represenation and fix the identity, which of course the automorphism group 
*must* do.

(Note, the ambient symmetric group these are embedded in is not Perm(G)
but rather S_{2n}, so one is tacitly identifying '1' with 'e_G' the identity
of G.)

	-Tim K.




On Tue, 19 Jul 2016, abdulhakeem alayiwola wrote:

> I am more interested in the procedure than the result. If anyone can help
> we the procedure or steps to find Aut(D2n) using GAP. I will be very glad
> on any hint as well.
> Regards
> 
> On Jul 18, 2016 4:56 PM, "abdulhakeem alayiwola" <lovepgroups at gmail.com>
> wrote:
> 
> > Dear forum,
> > Can anyone in the house describe the steps to find Aut(D2n) using GAP.
> > Note that Aut(D2n) is Automorphism group of Dihedral group of order 2n.
> > Regards.
> >
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
> 

--