[GAP Forum] Conjugacy classes Alternating group degree 125

[Hulpke,Alexander]

Dear GAP-Forum,

On Nov 26, 2016, at 8:17 AM, Víctor Manuel Ortiz Sotomayor <vicorso at doctor.upv.es<mailto:vicorso at doctor.upv.es>> wrote:

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?


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

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at colostate.edu<mailto:hulpke at colostate.edu>, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke