Question

When working with group cosets in MAGMA is there a way of treating the cosets as subsets of the overlying group. Specifically I have a group $G$ and subgroups $H$ and $K$ . I wish to look at the intersection of a pair of cosets $Hh$ and $Kk$ for some $h,k\in G$ , but am unable to perform such operations in MAGMA when they are considered as cosets.

Answer 1

As far as I can see, the only way to do that directly with cosets $ C1$ and $C2$ of $G$ is

$\{ x : x\ {\rm in}\ G\ |\ x\ {\rm in}\ C1\ {\rm and}\ x\ {\rm in}\ C2 \}$

which looks very inefficient, because it is iterating over all of $G$.

I would suggest first find a right transversal $T$ of $H \cap K$ in $H$, and then search through $T$ looking for an element $t \in T$ with $thk^{-1} \in K$. If you find such a $t$, then the intersection is the coset $(H \cap K)th$, and otherwise it is empty.

Answer 2

Well, this is trivial in GAP. Here is an example:

gap> G:=SymmetricGroup(7);; 
gap> H:=Stabilizer(G,1);;
gap> K:=SylowSubgroup(G,2);;
gap> c1:=RightCoset(H,(1,2));;  
gap> c2:=RightCoset(K,(1,2,3));;
gap> Intersection(c1,c2);      
[ (1,2,3), (1,2,3)(5,6), (1,2,3,4), (1,2,3,4)(5,6) ]
gap> 

By the way, GAP is free, unlike Magma...