[GAP Forum] to get correction in code.

[Stephen Linton]

Dear Harsha,

Your Autocenter function returns a list of the elements of the autocenter, but GAP distinguishes between a simple list of elements and a group.

The AsGroup operation will turn such a list into an actual group object (if the elements of the list form a group, as they do here).

gap> Autocenter(CyclicGroup(4));
[ <identity> of ..., f2 ]
gap> AsGroup(last);
<pc group of size 2 with 1 generators>
gap> StructureDescription(last);
“C2"

A couple of remarks that might be useful if you want to deal with larger groups:

There are certainly more efficient methods of computing the autocenter for larger groups. For instance you could start with the Centre of the group and then intersect it with the fixed subgroups of a set of generators for the outer automorphism groups.

Secondly, be aware of the limits of StructureDescription. For instance there are non-isomorphic groups with the same StructureDescription.


Finally, to construct a finitely presented group, follow the documentation at

http://www.gap-system.org/Manuals/doc/ref/chap47.html#X7AA982637E90B35A

	Steve

> On 28 Dec 2014, at 06:25, Harsha Arora <harshaarora.2008 at gmail.com> wrote:
> 
> Dear Sir
> 
> I have written code to get autocenter of a group , that is collection of
> all those elements in G which are fixed by all automorphisms of G, like
> center of G as centeral elements are fixed by all inner automorphisms of G
> . I have got the output. But i m not getting the structure description of
> the subgroup. I am getting the message no method is installed.   I also
> want to know the way to feed  a group with generators and relation.
> 
> Autocenter:=function(G)
> 
> local aut,auts,L;
> 
> aut:=AutomorphismGroup(G);
> 
> L:=Filtered(AsList(G),g->ForAll(AsList(aut), auts->Image(auts,g)=g));
> 
> return(L);
> 
> end;
> 
> 
> function( G ) ... end
> 
> Autocenter(CyclicGroup(4));
> 
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> 
> Error, no 1st choice method found for `AsList' on 2 arguments
> 
> Autocenter(CyclicGroup(4));
> 
> [ <identity> of ..., f2 ]
> 
> StructureDescription(Autocenter(CyclicGroup(4)));
> 
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> 
> Error, no 1st choice method found for `StructureDescription' on 1 arguments
> 
> IsSubgroup(CyclicGroup(4),Autocenter(CyclicGroup(4)));
> 
> false
> ALTERNATIVE  WAY IS
> 
> image:=function(G,g)
> 
> local L,aut,auts,i,j;
> 
> aut:=AutomorphismGroup(G);
> 
> L:=[];
> 
> for auts in aut do
> 
> AddSet(L, Image(auts,g));od;
> 
> return(L);
> 
> end;
> 
> function( G, g ) ... end
> 
> autc:=function(G)
> 
> local H;
> 
> H:=Filtered(G, g->Size(image(G,g)=1));
> 
> return(H);
> 
> end;
> 
> function( G ) ... end
> 
> autc(CyclicGroup(4));
> 
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> 
> Error, no 1st choice method found for `Size' on 1 arguments.
> 
> 
> PLEASE SOLVE MY PROBLEM.
> 
> With regards
> Harsha
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