[GAP Forum] listing all semidirect products contd.

[Jack Schmidt]

Using G=H=S3 as an example, here is the GAP code to do what you asked:

gap> LoadPackage("sonata");;
gap> g:=SymmetricGroup(3);; h:=SymmetricGroup(3);;
gap> ghs := List( Homomorphisms( g, AutomorphismGroup(h) ), f ->  
SemidirectProduct( g, f, h ) );
[ Group([ (7,8,9), (7,8), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (7,8,9), (3,6)(4,5)(7,8), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (7,8,9), (2,6)(4,5)(7,8), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (7,8,9), (2,3)(4,5)(7,8), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,3,6), (3,6)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,6,3), (3,6)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,6,3), (2,3)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,3,6), (2,3)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,3,6), (2,6)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]),
   Group([ (2,6,3), (2,6)(4,5), (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) ]
gap> List( ghs, IdGroup );
[ [ 36, 10 ], [ 36, 10 ], [ 36, 10 ], [ 36, 10 ], [ 36, 10 ], [ 36,  
10 ], [ 36, 10 ], [ 36, 10 ], [ 36, 10 ],
   [ 36, 10 ] ]


The main functions used are SymmetricGroup, Homomorphisms,  
AutomorphismGroup, SemidirectProduct, IdGroup, and List.  You can read  
the documentation for each within GAP using "?SymmetricGroup" etc. but  
most of them do just what you would guess.  IdGroup(a_group) returns  
[ size, serial_number ] where size is the order of the group and  
serial_number is a fixed (for all time) but (mostly) arbitrary number  
identifying the isomorphism type.  List( a_list, a_function ) returns  
the list you get by applying a_function to each element of a_list.  I  
use this to get a list of all the semidirect products from a list of  
all homomorphisms form G to Aut(H), and then again to get a list of  
the small group isomorphism type identification numbers.


On 2009-06-17, at 03:03, Dinesh Krithivasan wrote:

> Hi all,
>   This is a follow up to my earlier question regarding listing all  
> semidirect products of two groups. Thanks a lot for your response  
> Benjamin. I apologize to all for the triviality of the question. My  
> background is not group theory and I am trying to play with GAP to  
> better understand some of the concepts. What I am really looking for  
> is a way to find all possible semidirect products of G with H for  
> some arbitrary (but specified) groups G and H? For illustrative  
> purposes, I had chosen G and H to be S_3 in my previous mail which  
> turned out to be trivial. Thanks for the help and apologies for the  
> multiple mails.
>
> Regards,
> Dinesh.
>
>
>
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