> < ^ Date: Mon, 29 Nov 1999 11:26:29 +0100
> < ^ From: Stefan Kohl <kohl@mathematik.uni-stuttgart.de >
> < ^ Subject: Re: SmallGroup(12,1)

Dear Forum,

Kurt Ewald wrote:

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

This behaviour arises from the treatment of subgroups by GAP -
so GAP considers also, for example, the factors of a
direct product not as subgroups :

```gap> z3 := CyclicGroup(3);; z4 := CyclicGroup(4);;
gap> G := DirectProduct(z3,z4);;
gap> IsSubgroup(G,z3);
false
gap> IsSubgroup(G,z4);
false
```

Hope this helps,

Stefan

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