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> < ^ Subject:

Dear gap-forum,

Kurt Ewald wrote:

> Dear forum,

> s4:=SymmetricGroup(4)

> 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

> semidirectProduct(s,v)

> 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:

> G/H=Q

> 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.

Greetings,

Jan

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