[GAP Forum] Group automorphisms

[Alexander Hulpke]

Dear Forum, Dear Gabriel Bartolini,

>
> I'm not sure how to work with automorphisms in GAP.
The
AutomorphismGroup
of a (finite) group G is a group of homomorphisms.

> For example I would like to know if two vectors of elements in a  
> group G, (a,b,c) and (g,h,i)
> are equivalent under actions of automorphisms.
I suppose you consider your group as a direct product?
In any case, you could compute the
Orbit
of an element under the automorphism group.

For example, taking for G the symmetries of a square:

gap> G:=Group((1,2,3,4),(1,3));
Group([ (1,2,3,4), (1,3) ])
gap> A:=AutomorphismGroup(G);
<group of size 8 with 3 generators>
gap> Orbit(A,(1,3));
[ (1,3), (2,4), (1,4)(2,3), (1,2)(3,4) ]

this shows that (1,4)(2,3) and (1,3) are equivalent under the  
Automorphism group. If you do not want to search the whole orbit, you  
can also use

gap> RepresentativeAction(A,(1,3),(1,4)(2,3)); # example of equivalent  
elements
[ (1,4,3,2), (1,2)(3,4) ] -> [ (1,2,3,4), (2,4) ]
gap> RepresentativeAction(A,(1,3),(1,5));     # example of  
inequivalent elements
fail

Best,

    Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke