Dear Royce Peng,
You asked:
I'm having trouble creating semidirect products; I am trying to create all
semidirect products of a group of order 8 with a group of order 6. The
example in the manual invokes functions like InverseGeneralMapping and
IsomorphismPCGroup, but I don't understand their purpose.
For creating a semidirect product one needs groups G,H and a homomorphism
from G into the automorphism group of H. The functions
`InverseGeneralMapping' and `IsomorphismPCGroup' in the manual example
are used for creating this homomorphism. It is not necessary to use them.
The following example creates the semidirect product of S_3 with S_4,
where S_3 acts on S_4 by conjugation.
# First we create the groups. gap> G:= SymmetricGroup( 3 ); Sym( [ 1 .. 3 ] ) gap> H:= SymmetricGroup( 4 ); Sym( [ 1 .. 4 ] ) gap> eG:= Elements( G );; gap> AH:= AutomorphismGroup( H ); <group with 2 generators> # The we create the homomorphism by mapping each element `g' of `G' # onto the autmorphism `h -> h^g' of `H'. gap> imgs:= List( eG, g -> ConjugatorAutomorphism( H, g ) ); [ ^(), ^(2,3), ^(1,2), ^(1,2,3), ^(1,3,2), ^(1,3) ] gap> hom:= GroupHomomorphismByImages( G, AH, eG, imgs ); [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ] -> [ ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), () ), ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), (2,3) ), ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), (1,2) ), ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), (1,2,3) ), ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), (1,3,2) ), ConjugatorAutomorphism( SymmetricGroup( [ 1 .. 4 ] ), (1,3) ) ]
# Finally we create the semidirect product.
gap> P:= SemidirectProduct( G, hom, H );
<permutation group with 4 generators>
I hope this is of help. Sorry for answering so late.
Best wishes,
Willem de Graaf
School of Mathematical and Computational Sciences University of St Andrews North Haugh Tel: +44 1334 463273 St Andrews Email: wdg@dcs.st-and.ac.uk Fife KY16 9SS Scotland