[GAP Forum] efficient ways of computing orbit-stabilizers on equivalence classes of surjections
Will Chen
oxeimon at gmail.com
Fri Sep 11 08:55:12 BST 2015
Let me preface this by saying that my version of gap is rather dated
(version 4.6.4), though I do have a newer build which I had to compile with
the option "./configure --with-gmp=no"
Anyway, my problem is this:
Let F2 be the free group on two generators, and let G be a finite group
(I'm mostly interested in the case where G is simple)
Let X be the set of equivalence classes of surjective homomorphisms F2 ->
G, modulo conjugation in G.
There's a natural action of the SL(2,Z) on X, acting as outer automorphisms
of F2.
I want to compute everything about this action, namely the orbits, and the
stabilizers.
My current routine essentially goes like this:
1. Compute GQuotients(F2,G)
2. Compute the orbits of Aut(G) on GQuotients(F2,G) to get a complete list
of surjections F2 -> G, call it GQComplete
3. Compute the orbits of Inn(G) on GQComplete, store each orbit
"AsSSortedList" (strictly sorted list)
4. Let SAut(F2) be the subgroup of Aut(F2) mapping onto SL(2,Z) (ie,
automorphisms of "determinant 1"). Then I compute the orbits of SAut(F2)
acting by composition on the set of orbits from (3), where each orbit is a
"strictly sorted list".
5. Pick representatives of each orbit, compute their stabilizers.
Is this basically the best one can do? In particular, are there better
alternatives to strictly sorted lists for computing actions on equivalence
classes of this form? Are there better ways of getting all surjections
immediately without the Aut(G)-orbit computation?
So far the largest group my computer seems to be able to handle with my
current routine without running out of memory is PSL(2,7) of size 168.
Any ideas?
thanks
- will
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