[GAP Forum] Product of Subgroups in a Group

[Burkhard Höfling]

On 2013-02-15, at 10:35 , Sven Reichard <Sven.Reichard at tu-dresden.de> wrote:

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> Am 15.02.2013 10:02, schrieb rahul kitture:
>> Given two subgroups $H$ and $K$ of a finite group (say Symmetric/ 
>> Alternating Group), how do we compute the product $HK$ in the
>> group? I couldn't find anything from Help or topics in online
>> library.
> 
> This may not be the most elegant way, but
> gap> Group(Concatenation(List([H,K], GeneratorsOfGroup)), Identity(H));
> should do the trick.


This gives the subgroup generated by H and K but not, in general, the set HK. If you know that HK is a subgroup, then this will work. ClosureGroup (H,K) might be a bit more efficient.

On 2013-02-15, at 14:27 , Sandeep Murthy <sandeepr.murthy at gmail.com> wrote:

> Hi,
> 
> If H, K are subgroups of G then
> 
> ListX( H, K, PROD )
> 
> will return a (mutable) list of
> the elements
> of the set HK in
> G.

But note that the list will have duplicates if H and K don't intersect trivially.

I would suggest DoubleCoset (H, One(H), K)  to represent HK more efficiently. You can turn the double coset into a list via AsList/ AsSSortedList.

Cheers

Burkhard.