Dear gap-forum,

In answer to Kurt Ewald's question (see beelow):

Given two groups G and H in GAP, there is a difference between the
statement that H is a subgroup of G, and the statement that G has
a subgroup isomorphic to H. To find z4 in your p, you can use
Embedding(p,1);
this gives a monomorphism z4->p. To find z3, you can use
Embedding(p,2);
To find the projection of p onto z4, use
Projection(p);

Best wishes,

Jan
> Dear forum
> theory says that SmallGroup(12,1) is the semidirect product of z3 by z4
> indeed
> gap> z3;z4;
> Group([ (1,2,3) ])
> Group([ (1,2,3,4) ])
> gap> aut:=AutomorphismGroup(z3);
> <group of size 2 with 1 generators>
>  hom:=GroupHomomorphismByImages(z4,aut,[(1,2,3,4)],[aut.1]);
> gap> p:=SemidirectProduct(z4,hom,z3);
> Group([ (2,3)(4,5,6,7), (1,2,3) ])
> gap> x:=SmallGroup(12,1);
> <pc group of size 12 with 3 generators>
> gap> IsomorphismGroups(p,x);
> [ (2,3)(4,5,6,7), (1,3,2) ] -> [ f1, f3 ]
> But
> gap> IsSubgroup(p,z4);
> false
> gap> IsSubgroup(x,z4);
> false
> The complement (z4) of z3 must be a subgroup of the semiproduct.
> Where lies the error?
> Best wishes
> K. Ewald
> 
> 

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