[GAP Forum] Generating a group from a triple of elements.

[johnathon simons]

Hi everyone,

I'm interested in finding rigid triples for simple groups – Let G be a simple group and g_i in a conjugacy class C_i of G. We say that G is rationally rigid if:
1)G = <g_1, g_2, g_3>
2)g_1*g_2*g_3 = 1
3)http://www.maths.qmul.ac.uk/~raw/pubs_files/sgensweb.pdf (page 3 provides the condition of the “symmetrised structure constant” and it being = 1).
Standard generators for sporadic simple groups - QMUL Maths<http://www.maths.qmul.ac.uk/~raw/pubs_files/sgensweb.pdf>
www.maths.qmul.ac.uk
Standard generators for sporadic simple groups Robert A. Wilson School of Mathematics and Statistics, The University of Birmingham, Edgbaston, Birmingham B15 2TT



If one looks to page 4 of the above document it provides the same definition of rational rigid generators of the group.

So far, I have a method of plugging in arbitrary triples of conjugacy classes to verify whether a group G can be expressed as a random triple of elements from a conjugacy class (see below).


“findNiceTriple := function(G, cls1, cls2, cls3)
    local g1, g2, g3;
    g1 := Representative(cls1);
    for g2 in cls2 do
        g3 := (g1*g2)^-1;
        if g3 in cls3 and M11 = Group(g1, g2) then
            return [g1, g2, g3];
        fi;
    od;
    return fail;
end;

Then for example for the Mathieu simple group (M11):

gap> M11:=MathieuGroup(11);
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])
gap> rc:=RationalClasses(M11);;
gap> Length(rc);
8
gap> findNiceTriple(M11, rc[2], rc[5], rc[8]);
[ (1,8)(3,9)(5,7)(10,11), (1,7,6,3,4,2,11,9,5,8,10), (1,11,2,4,9,10)(3,6,5)(7,8) ]
gap> findNiceTriple(M11, rc[8], rc[8], rc[8]);
[ (1,3,4,9,5,11)(2,6,10)(7,8), (1,4,7,5,10,9)(2,6,11)(3,8), (1,4,11,2,5,8)(3,7)(6,10,9) ]”

Questions:

Could someone please direct me towards a method of also implementing the third condition (that of the “symmetrized structure constant” being equal to 1 – if the code could some how calculate the value of the constant so to realize whether the group can be seen as rationally rigid).

Secondly, the above works for random rational classes, but in the literature, conjugacy classes are written in ATLAS notation (e.g 2A, 2B which signify the order of the elements in the class are 2 and according to http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/doc/manual.pdf the alphabet signifies a decreasing cenralizer order). Could someone also help me so that the code focusses only on rational conjugacy classes (rational classes are implemented on GAP) in ATLAS notation because if I’m not mistaken there is a difference between the class 2A/2B which I’m not sure of how to differentiate in the notation “rc[2]”.
AtlasRep A GAP 4 Package - math.rwth-aachen.de<http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/doc/manual.pdf>
www.math.rwth-aachen.de
AtlasRep — A GAP 4 Package (Version 1.5.1) Robert A. Wilson Richard A. Parker Simon Nickerson John N. Bray Thomas Breuer Robert A. Wilson Email: R.A.Wilson at qmul.ac.uk



Essentially, I just want to check whether the group M11 has rationally rigid generators and this appears to be the most standard method. If any one knows of any simpler method I would be more than appreciative.

With the deepest regards,

John