Kurt Ewald wrote:
> Dear forum,
> gap> t;
> Group([ (1,4,3), (1,4) ]) is a subgroup of s4, but not normal
> gap> v;
> Group([ (1,2)(3,4), (1,3)(2,4) ]) is a normalsubgroup of s4
> Because t is a complement to v in s4, s4 is a semidirectproduct of t by v.
> But using the automorphismGroup of t all the
> GroupHomomorphismByImages failed and the construction of the
> was impossinble.
> Are there other methods?
I don't understand your question. Perhaps this is what you want to do:
t:=Group([ (1,4,3), (1,4) ]); v:=Group([ (1,2)(3,4), (1,3)(2,4) ]); hom:=GroupHomomorphismByFunction(t,AutomorphismGroup(v), a->ConjugatorAutomorphism(v,a)); g:=SemidirectProduct(t,hom,v);
and this works. In fact, the so constructed g turns out to be _equal_
to SymmetricGroup(4) in GAP.
In fact, I have a related question: is there a SemidirectProduct
function for Lie algebras in GAP? Or did someone make it already? That
would save me some work..
> A second question:
> knowing H and Q can GAP construct G or the isomorphismClass of G?
The isomorphism class of G is by no means unique from this equation,
and to compute all possibilities is in general a hard problem. I do
not know the details in the case of groups, but for Lie algebras, in
case H is an Abelian ideal, the possible extensions are parametrized
by the 2-cohomology group H^2(G/H,H). Something similar should hold
for groups (for an Abelian H?), and I guess you can do some cohomology
computations in GAP.