[GAP Forum] Group automorphisms
Alexander Hulpke
hulpke at math.colostate.edu
Tue May 26 16:50:31 BST 2009
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
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