[GAP Forum] perfect residuum?

[Hulpke,Alexander]

Dear Forum, Dear Keith Dennis,

When calculating maximal subgroups, there is a reduction to the case of (almost) simple groups (you are already in this base case) andthere are three ways to deal with such groups:

a) Pretabulated information (e.g. for sporadic groups). This will always be limited

b) Explicit listing of maximal subgroups following e.g the book by Bray/Holt/Roney-Dougal. This is (alas) not yet implemented in GAP, though it clearly is the method to use. The issue is not that this is fundamentally hard to implement, but that it requires care to ensure the manifold case distinctions do not miss any case.

c) Calculate the subgroup lattice from scratch and take the resulting maximal subgroups. This calculation requires perfect subgroups as seed. The error you get is because the library of perfect groups (which is used to supply such information) does not go far enough. In the specific situation of PSL_2 (for which the only perfect subgroups are A or 2.A5 and PSL_2's for subfields, it can easily be that the data base actually suffices -- this is what you observed. However, had you used PSL(2,17^4) and returned after the error, this would have missed out on PSL(2,17^2) (since it has order >10^6) andthus  given a wrong result.

> 2.  Such interruptions occur for other groups such as PSL(2,p) for
> smaller values.  Now one knows what the maximal subgroups are for
> PSL(2,p), as well as PSL(2,q), q=p^n, at least up to isomorphism.  One
> can use this plus IsomorphicSubgroups(g,M) to find the maximal
> subgroups, but that doesn't seem to be very efficient.
> 
> Is there a way to do this directly? 

You probably want to go back to the pre-image SL_2(q), then the tables on p. 377 give you how to find the subgroups (that is exactly what point b) above would do). It is just a bit beyond the limit what I would be offering to code ad-hoc, when answering a question.

All the 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, 
http://www.math.colostate.edu/~hulpke