[GAP Forum] GAP Help to compute size of permutations

[James Mitchell]

Dear Shubh, 

If you want to know the size of the group of units of your transformation semigroup, and you have the Semigroups package loaded, then you can simply type GroupOfUnits(S); where S is your transformation semigroup. If you don't want the group of units but only those permutations in S, then you can do the same thing, and just check if the elements of the GroupOfUnits are permutations (check DegreeOfTransformation(f) = RankOfTransformation(f) where f = Representative(GroupOfUnits(S))). If this is true, then every element of the group of units is a permutation, if this is false, then there are no permutations in S. 

Maybe this is not what you wanted, if not, then please elaborate what "all permutations" should mean in this context, as I can think of at least 3 different possible meanings (2 of which are the above, the third is that you want to know the set of all the transformations in S that act as permutations on their images, which you can obtain by doing List(Idempotents(S), x -> HClass(S, x)); again with the Semigroups package loaded).

Best wishes,

James

James Mitchell
jdm3 at st-andrews.ac.uk



> On 28 Oct 2019, at 04:36, Shubh Narayan Singh (Assistant Professor, Mathematics) <shubh at cub.ac.in> wrote:
> 
> Dear all,
> 
> Can anybody tell me a GAP code, if any, to compute the size of the set of
> all permutations in a transformation semigroup.
> 
> Thanks in advance for your assistance.
> 
> -- 
> 
> Sincerely,
> *------------------------------------------*
> 
> *Shubh N. Singh, Ph.D.Assistant Professor *
> 
> *Department of MathematicsCentral University of South Bihar*
> 
> *Gaya–824236 (Bihar), INDIA-----------------------------------------*
> *E-mail: shubh at cub.ac.in <shubh at cub.ac.in>*
> *Mob:   +91- 8404934146*
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