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

[Thomas Breuer]

Dear GAP Forum,

Jonathon Simons asked about a way to compute
symmetrized structure constants in GAP.

The function 'ClassStructureCharTable' does almost
what one wants.
In the case of triples, the result of that function
has to be divided by the group order.

In the example of the Mathieu group M11,
this could look as follows.
(For the sake of simplicity, I am using the character table
from GAP's library.)

    gap> t:= CharacterTable( "M11" );
    CharacterTable( "M11" )
    gap> Size( t );
    7920
    gap> OrdersClassRepresentatives( t );
    [ 1, 2, 3, 4, 5, 6, 8, 8, 11, 11 ]
    gap> ClassStructureCharTable( t, [ 2, 4, 9 ] ) / Size( t );
    1

This means that the symmetrized structure constant
for a triple of classes of element orders 2, 4, and 11
has the value 1.

I am not sure whether I understand the second question,
about rational conjugacy classes.
The class names in ATLAS notation (2A, 2B, ...) refer always
to conjugacy classes, not to rational classes.
If one has the set of conjugacy classes and wants to decide
whether the G-conjugacy class of the element g is rational
then one can check whether all generators of the cyclic group
generated by g are conjugate in G.
(If one knows the character table of G then one could also
check whether all irreducible characters are rational at g.)

All the best,
Thomas



On Thu, Aug 17, 2017 at 11:10:14AM +0000, johnathon simons wrote:
> 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