[GAP Forum] Creating Sp(4, Z/2^nZ)

[Hulpke,Alexander]

Dear Forum, Dear Watson Ladd,

> I am interested in a problem about the lifting of mod
> p-representations to p-adic representations. To do this I want to
> construct the group Sp(4, Z/2^nZ) for some small
> values of n in GAP along with the reduction homomorphisms.

For example:

gap> g:=SP(4,Integers mod 32);
Sp(4,Z/32Z)
gap> gens:=GeneratorsOfGroup(g);;
gap> geni:=List(gens,m->List(m,r->List(r,Int)));;  # generators as Z matrices
gap> geni:=List(gens,m->List(m,r->List(r,Int)))*Z(2);  # move to GF(2)
gap> h:=Group(geni);
<matrix group with 6 generators>
gap> hom:=GroupHomomorphismByImagesNC(g,h,gens,geni);

(note that the last command will take a bit.)


> If I knew generators this would be easy, but I don't. Does anyone have
> ideas for what to do? Also, if there are any books I should look at
> for the theory that would be useful: I don't know much about this area
> yet as I got thrust into it by research demands.

May I plug my paper ``Computing generators of groups preserving a bilinear form over residue class rings’’, J.Symb.Comp, 50 (2013), 298-307. DOI  10.1016/j.jsc.2012.08.002

Concerning the structure of these groups, you also might find theorem 2.5 in the joint paper
https://arxiv.org/abs/1611.05921
(to appear in Math.Comp.) useful.

Best wishes,

  Alexander Hulpke

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