[GAP Forum] Product of Subgroups in a Group

[William DeMeo]

See also the thread "complex product of two groups" in the forum archive:

http://mail.gap-system.org/pipermail/forum/2012/thread.html#3628

Cheers,
William



On Fri, Feb 15, 2013 at 10:51 AM, Benjamin Sambale
<benjamin.sambale at gmail.com> wrote:
> See also SetX(H,K,PROD)
>
> Benjamin
>
> Am 15.02.2013 15:47, schrieb 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:
>>>
>>>> -----BEGIN PGP SIGNED MESSAGE-----
>>>> Hash: SHA1
>>>>
>>>> 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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