[GAP Forum] normal subgroup semidirect product

[Alexander Hulpke]

Dear Bill/Esther, Dear Forum,

> I need to know the normal subgroups of the following group G'.
> 
> G' is defined as follows.
> 
> G=symmetric group on 7 elements.
> 
> S=symmetric group on 5 elements, embedded in G as the subgroup stabilizing 6
> and 7.
> 
> R = integers mod 3.
> 
> K=the free R-module with R-basis the set of right cosets of S in G.
> 
> Finally G' is the semidirect product of K by G.

I read this description that (n=[S_7:S_5]=7*6=42) you have S_7 act in degree 42 on pairs of points and take the semidirect product of K=C_3^n with S_7 acting by permuting generators. The main issue is how to generate this group, as the generic semidirect product, or the product generated by matrices are rather unhandy to work with.

The easiest way in fact seems to be to observe that (because of the permutation of the basis vectors) the group is in fact simply the wreath product C_3\wr S_7 with S_7 acting on 42 points:

gap> S:=SymmetricGroup(7);
Sym( [ 1 .. 7 ] )
gap> pairs:=Arrangements([1..7],2);
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 1 ], 
  [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 1 ], [ 3, 2 ], 
  [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 3, 7 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ], 
  [ 4, 5 ], [ 4, 6 ], [ 4, 7 ], [ 5, 1 ], [ 5, 2 ], [ 5, 3 ], [ 5, 4 ], 
  [ 5, 6 ], [ 5, 7 ], [ 6, 1 ], [ 6, 2 ], [ 6, 3 ], [ 6, 4 ], [ 6, 5 ], 
  [ 6, 7 ], [ 7, 1 ], [ 7, 2 ], [ 7, 3 ], [ 7, 4 ], [ 7, 5 ], [ 7, 6 ] ]
gap> hom:=ActionHomomorphism(S,pairs,OnTuples);
<action homomorphism>
gap> G:=Image(hom);
Group([ (1,8,15,22,29,36,37)(2,9,16,23,30,31,38)(3,10,17,24,25,32,39)(4,11,18,
    19,26,33,40)(5,12,13,20,27,34,41)(6,7,14,21,28,35,42), 
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,14)(19,20)(25,26)(31,32)(37,38) ])
gap> Size(G);
5040

gap> P:=WreathProduct(Group((1,2,3)),G);
<permutation group of size 551471705222822290413360 with 44 generators>

Now we can calculate the normal subgroups (32 in total)
gap> N:=NormalSubgroups(P);

Hope this helps,

   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