[GAP Forum] Finding [2,p]-generated groups

[Alexander Hulpke]

Dear Keith, Dear Forum,

On Jun 15, 2007, at 6/15/07 8:44, R. Keith Dennis wrote:

> I'm doing similar, but more complicated computations now.  In your
> case you can eliminate many groups from consideration almost instantly
> by using EulerianFunction(G,2) to determine if the group is generated
> by 2 elements or not.  After that, I suppose you could save a bit
> by letting the first element range over representatives of conjugacy
> classes of elements of order 2 (resp p), but for the second element
> testing all relevant ones seems like the simplest way to go.
It is possible to select the second element up to conjugacy with the  
centralizer of the first element. (A nice description is in section  
9.1 of Holt, Eick, O'Brien: Handbook of CGT).

The easiest way to do this in GAP is to use the function GQuotients  
on a suitable finitely presented group:
For example one case for p=11:

gap> f:=FreeGroup("x","y");
<free group on the generators [ x, y ]>
gap> AssignGeneratorVariables(f);
#I  Assigned the global variables [ x, y ]
gap> rels:=[x^2,y^11]; #e.g. p=11
[ x^2, y^11 ]
gap> g:=f/rels;
<fp group on the generators [ x, y ]>
gap> h:=MathieuGroup(11);
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])
gap> GQuotients(g,h);
[ [ x, y ] -> [ (4,10)(5,8)(6,7)(9,11), (1,6,2,10,3,4,8,7,11,9,5) ],
   ...

This lists one representative per conjugacy class of generating  
systems. The list returned is empty iff no generating system exists.

I hope this is of help,

      Alexander