[GAP Forum] Semidirect products

[Alexander Hulpke]

Dear GAP Forum,

On Oct 4, 2011, at 10/4/11 1:28, Sandeep Murthy wrote:

> 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.

Here are a couple of products and thus multiple steps of of embeddings, but I think being just really stubborn with the definitions produces the desired result. Of course there are better ways to represent G (e.g. as PC group), but you get the desired subset. For example:

C:=Group((1,2,3,4,5));
H:=DirectProduct(C,C,C);
Hemb:=List([1..3],x->Embedding(H,x));
Hi:=List(Hemb,x->Image(x,C)); #H1,2,3

HS:=DirectProduct(H,H); #H^2
HSE:=List([1,2],x->Embedding(HS,x));

# construct swapper by finding generators in both components
Hgens:=List(HSE,x->List(GeneratorsOfGroup(H),y->Image(x,y)));

# homomorphism swapping both copies
swapper:=GroupHomomorphismByImages(HS,HS,
  Concatenation(Hgens[1],Hgens[2]),
  Concatenation(Hgens[2],Hgens[1]));

U:=Group((1,2));

G:=SemidirectProduct(U,
    GroupHomomorphismByImages(U,Group(swapper),[U.1],[swapper]),HS);

Uemb:=Embedding(G,1);
HSemb:=Embedding(G,2);


#Now produce the three slices of S first, S is the set of all products.

# elements (a,1,1) with a<>1
Scomp1:=List(Difference(Hi[1],[One(Hi[1])]),x->Image(HSemb,Image(HSE[1],x)));
# elements (1,b,1)
Scomp2:=List(Hi[2],x->Image(HSemb,Image(HSE[2],x)));
# elements (1,1,j)
Scomp3:=Elements(Image(Uemb));

S:=List(Cartesian(Scomp1,Scomp2,Scomp3),x->x[1]*x[2]*x[3]);

T and U work analogously.

Best,

   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