[GAP Forum] Product of Subgroups in a Group

[Sandeep Murthy]

the duplicates can be removed using

Unique( ListX( H, K, PROD ) ).

Seems fairly quick, but maybe for large products
DoubleCoset (H, One(H), K), suggested by Burkhard,
is quicker.

Sincerely, Sandeep.

Burkhard Höfling wrote:
> 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.
>
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