[GAP Forum] Direct Product Group in GAP

[Sergio Siccha]

Dear Lopo Apelo Kosho,

I guess the problem is due to the facts that:
(a) `DirectProduct` chooses an "efficient representation", essentially
ignoring all fixed points. For example as here:

gap> G := Group((1,2,4));
Group([ (1,2,4) ])
gap> DirectProduct(G, G);
Group([ (1,2,3), (4,5,6) ])

(b) GAP identifies all permutations and hence also permutation groups
which move the same points in the same way, that is which differ only by
fixed points. So to GAP the identity subgroups of A_4 and S_3 are identical.

Now I create D, w1, and w2 as you described:

gap> A4 := AlternatingGroup(4);; S3 := SymmetricGroup(3);;
gap> D := DirectProduct(A4, S3);
Group([ (1,2,3), (2,3,4), (5,6,7), (5,6) ])
gap> w1 := Group(());; w2 := Group((1,2,3));;
gap> DirectProduct(w1, w2);
Group([ (), (1,2,3) ])

Then `DirectProduct(w1, w2)` is a subgroup of A_4, which is why GAP also
states that it is not normal in D.

Since there is no way to tell GAP that something should be the trivial
group acting on the set {1,2,3,4}, we can't tell it that w1 should
actually fix {1,2,3,4}. However, if you look into the documentation of
`DirectProduct` you find examples of how to use the functions
`Embedding`, `Projection` and `Image` to do what I think you want to do.

Hope to help,
Sergio

On 22.06.21 07:58, lopo apelo kosho wrote:
> Dear Friends, 
> I am writing a GAP code to find whether certain subgroup properties are preserved under direct products in finite groups.  But it seems that Direct product of groups in GAP has some different setting.  
> For instance, If we D= DirectProduct(A_4,S_3); where A_4 is the alternating group of order 12 and S_3 is the symmetric group in three letters; and if 
> w1=the identity subgroup in A_4 
> w2=the subgroup of order 3 in S3
> W=DirectProduct(w1,w2)
> Then GAP recognize W as subgroup in D, GAP recognize W as non-normal subgroup. What is wrong here? and how get the elements of the direct product group as ordered pairs in GAP. 
> Thank you. 
> 
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