[GAP Forum] split extensions

[Alexander Hulpke]

Dear GAP Forum,

Alex Trofimuk asked:

> How to find a split extensions G of group H by group N, i.e. G/N is  
> isomorphic to H?
As you defined N and H independently, you presumably want to construct  
a semidirect product. For this, you would have to find a homomorphism  
from H into the automorphism group of N. Then `SemidirectProduct' will  
construct the extension (see manual).

In your concrete example it turns out that the largest subgroup of Aut 
(N) which is isomorphic to a factor of H has order 6, so for example  
the following is a brute-force way to construct such a product:

gap> a:=AutomorphismGroup(N);
<group of size 12000 with 2 generators>
gap> u:=ConjugacyClassesSubgroups(a);;
gap> u:=List(u,Representative);;
gap> u6:=Filtered(u,x->Size(x)=6);
[ <group of size 6 with 2 generators>, <group of size 6 with 2  
generators> ]
gap> GQuotients(H,u6[2]);
[ [ a, b, c ] -> [ Pcgs([ f1, f2, f3 ]) -> [ f2^3, f1^2, f3^4 ],
       [ f1^4*f2^2*f3, f1^2, f3 ] -> Pcgs([ f1, f2, f3 ]),
       Pcgs([ f1, f2, f3 ]) -> [ f1^4*f2^2*f3, f2, f3^4 ] ] ]
gap> hom:=last[1];
[ a, b, c ] -> [ Pcgs([ f1, f2, f3 ]) -> [ f2^3, f1^2, f3^4 ],
   [ f1^4*f2^2*f3, f1^2, f3 ] -> Pcgs([ f1, f2, f3 ]),
   Pcgs([ f1, f2, f3 ]) -> [ f1^4*f2^2*f3, f2, f3^4 ] ]
(If you know the homomorphism, you would not need to go through all  
this brute-forcing)

gap> sdp:=SemidirectProduct(H,hom,N);
<pc group with 8 generators>
gap> Size(sdp);
6000

See the manual for connections between the original groups and the new  
product.

Regards,

     Alexander Hulpke


>
> So I define group H:
>
> gap> F:=FreeGroup("a","b","c");
> <free group on the generators [ a, b, c ]>
> gap> a:=F.1;b:=F.2;c:=F.3;
> a
> b
> c
> gap> rels:=[a*(b*c)^-1,b^3*(a*b*c)^-1,a*b*(c^-1)^3];
> [ a*c^-1*b^-1, b^3*c^-1*b^-1*a^-1, a*b*c^-3 ]
> gap> H:=F/rels;
> <fp group on the generators [ a, b, c ]>
> gap> Size(H);
> 48
>
> So I define group N:
> N:=ExtraspecialGroup(125,5);
> <pc group of size 125 with 3 generators>



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