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>
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hul...@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
