[GAP Forum] Conjugacy classes Alternating group degree 125

[VMOS]

First of all, thank you so much por your invaluable help. I'm a young student and I didn't know that "the conjugacy classes of the alternating groups were well-known and there sizes could be computed easily". Anyway, I hope that this kind of computations with many conjugacy classes might be interesting in other frameworks.

I'm sorry if the ease of my question has bothered someone.


Best,

VMOS

<div>-------- Mensaje original --------</div><div>De: Víctor Manuel Ortiz Sotomayor <vicorso at doctor.upv.es> </div><div>Fecha:27/11/2016  15:44  (GMT+01:00) </div><div>Para: benjamin.sambale at gmail.com,Alexander.Hulpke at colostate.edu </div><div>Cc: Forum at mail.gap-system.org </div><div>Asunto: [GAP Forum] Conjugacy classes Alternating group degree 125 </div><div>
</div>



-------------------------------------------------------------------------

> My recommendation would be to take the code for computing conjugacy  
> classes (which is in the file `claspcgs.gi`) and centralizers (which  
> works down over subsequently larger factor groups) which currently  
> works breadth-first to convert it to a depth-first approach, not  
> storing all classes, but deleting them once the size of the class is  
> known.

> Caveat: Even the factor of Q modulo the 6th term in the lower  
> central series (this is less than sqrt(|Q|), so only a small bit)  
> already has almost 2 million conjugacy classes, so the whole group  
> could easily have 10^10 classes or more. which puts the feasibility  
> of such an enumerative approach into doubt.

> Best,

>  Alexander Hulpke


--------------------------------------------------------------------------

>> If I understand the question correctly, there is no need to use a  
>> computer. The conjugacy classes of the alternating groups are  
>> well-known and there sizes can be computed easily. But maybe this  
>> is not the point of the question.

>> Best,
>> Benjamin

--------------------------------------------------------------------------

>>> Am 26.11.2016 um 16:17 schrieb Víctor Manuel Ortiz Sotomayor:
>>> Let G:=AlternatingGroup(125) be the Alternating group of degree  
>>> 125, and let Q:=SylowSubgroup(G, 5) be a Sylow 5-subgroup of G.
>>>
>>> I want to compute, for each element x of Q, the distinct  
>>> G-conjugacy class sizes, that is, the distinct values of  
>>> Size(ConjugacyClass(G, x)) (obviously, computing the distinct  
>>> values of Centralizer(G, x) for all x in Q) would be the same).
>>>
>>> Needless to say that, I always get out of memory when I run over  
>>> all the elements of Q. I had tried the following: compute the  
>>> upper central series of Q (L:=UpperCentralSeriesOfGroup(Q)) and,  
>>> for some "intermediate" normal subgroup N in that chain, to  
>>> decompose Q in right cosets on N, in order to make a disjoint  
>>> union of the elements of Q that is more manageable. However, I  
>>> still have problems of memory because either I have so many  
>>> transversals or the order of N is also too large. Any idea?
>>>
>>> Thanks in advance.