[GAP Forum] Re: Sampling elements in conjugacy classes of a symmetric group

[Alexander Hulpke]

Don King wrote:

> I`d like to sample some elements in a symmetric group of order n  
> based on the ratio of conjugacy classes.
>
> For instance, if a symmetric group just has five conjugacy classes  
> (this is just for illustration, not an actual symmetric group),
>
> Class 1: 10 memeber
> Class2: 20 members
> Class3: 30 memebers
> Class4: 20 members
> Class5: 10 memebers
>
> is it possible to pick random samples like 1,2,3,2,1 elements from  
> each class based on the ratio of each conjugacy classes ?.
>
> Is there any way to pick random samples from each conjugacy class ?.

If you take all classes of G, this of course is just random sampling  
over the whole of G, and
Random(G)
in GAP will do this. If you only have a subset of classes in the list  
class, the following commands will produce such weighted random  
sampling:

Precompute:
sz:=List(class,sz);
gcd:=Gcd(sz);
sz:=List(sz,x->x/gcd);
sums:=List([1..Length(sz)],x->Sum(sz{[1..x-1]}));
m:=Sum(sz);
Add(sz,m);

Sampling:
r:=Random([1..m]);
p:=First([1..Length(class)],x->sz[x]<r and sz[x+1]>=r);
randelm:=Random(class[p]);   # random group element


Best,

   Alexander Hulpke



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