[GAP Forum] Semidirect products

[Sandeep Murthy]

Thanks.

I've got one concrete problem for a semidirect product on GAP
which I am having problems with.

It is to get  a handle on three subsets of the semidirect product group

G := H^2 \rtimes C_2

where H := C_5^3 (3-fold direct product of the cyclic group of 5 elements),
and C_2 acts on H^2 by switching its two factors.  The group H^2 is of
order 15625, and G is order 31250.  Elements of G can be written as
(a,b)z, where a, b \in H, and z is the generator of C_2.

If 1 is the identity of C_5, and

      H_1 := C_5 \times {1} \times {1}
      H_2 := {1} \times C_5 \times {1}
      H_3 := {1} \times {1} \times C_5

are the three subgroups of H isomorphic with the three copies
of C_5, then how can I get a handle on the following subsets S_1, S_2, S_3
with the definitions:

S := { (a,b)z^j  |  a \in H_1 \ {(1,1,1)},  b \in H_2, j \in {0,1} },
T := { (c,d)z^j  |  c \in H_2 \ {(1,1,1)},  d \in H_3, j \in {0,1} },
U := { (e,f)z^j  | e \in H_3 \ {(1,1,1)},  f \in H_1, j \in {0,1} }.

I tried to get the subgroups H_i as images of appropriate embeddings
of C5 in G, via intermediate embeddings.  But I am not getting the right
images.  S, T, U should each be of size 40.

Sincerely, Sandeep.





On 4 Oct 2011, at 16:16, Alexander Hulpke wrote:

> 
> 
> Dear GAP-Forum,
> 
> On Oct 4, 2011, at 10/4/11 8:57, Sandeep Murthy wrote:
>> is there a quick way to directly access the factors of a semidirect product group?
>> I have constructed a semidirect product G = N \rtimes_\theta P
> 
> According to the manual, section 47.2 (Semidirect product):
> 
> Embedding(G,1) returns the embedding P->G, Embedding(G,2) that of N. The subgroups of G you want then can be obtained as Image of these maps. For example:
> 
> gap> G:=SemidirectProduct(GL(3,2),GF(2)^3);
> <matrix group of size 1344 with 3 generators>
> gap> hom1:=Embedding(G,1);
> CompositionMapping( [ (5,7)(6,8), (2,3,5)(4,7,6) ] -> 
> [ <an immutable 4x4 matrix over GF2>, <an immutable 4x4 matrix over GF2> 
> ], <action isomorphism> )
> gap> Pimg:=Image(hom1);
> <matrix group of size 168 with 2 generators>
> gap> Size(Pimg);
> 168
> gap> hom2:=Embedding(G,2);
> MappingByFunction( ( GF(2)^3 ), <matrix group with 
> 3 generators>, function( v ) ... end, function( a ) ... end )
> gap> Nimg:=Image(hom2);
> <matrix group of size 8 with 3 generators>
> gap> Size(Nimg);
> 8
> 
> Regards,
> 
>  Alexander Hulpke
> 
> 
> 
> -- Colorado State University, Department of Mathematics,
> Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
> http://www.math.colostate.edu/~hulpke
> 
> 
>