[GAP Forum] Generating Set for Special Linear Groups

[Stephen Linton]

Dear GAP Forum,

On 7 Jun 2011, at 01:56, Piyush garg wrote:

> I am very tahnkful to Peter, Stefan and Steve for helping me out.
> Now I am able to do this through GeneratorsOfGroup and SmallGeneratingSet ,
> but I am still getting the error message for MinimalGeneratingSet (thats ok,
> i will keep it on hold for moment).

As I said, the next version will give a better error message, but we don't have a way to 
compute a minimal cardinality generating set efficiently for non-solvable groups.
> 
> I have two more doubts :
> ##    How to get the matrices from the output generated by GeneratorsOfGroup
> for the Special Linear Group;  like for SL(2,17) :
> s:=SL(2,17);
> gens:=GeneratorsOfGroup(s);
> [ [ [ Z(17), 0*Z(17)], [ 0*Z(17), Z(17)^15 ] ],
>  [ [ Z(17)^8, Z(17)^0], [Z(17)^8, 0*Z(17)] ] ]
> 
> this is the output we are getting, now hwo to get the matrices from it which
> can generate the whole set of SL(2,17) ??  (with the elements' values from 0
> to 16.)
> 

Well, this is two matrices that generate the whole of SL(2,17), but not represented in the way you want.
You can visualize them in the form you want using Display

gap> Display(gens[1]);
  3  .
  .  6
gap> Display(gens[2]);
 16  1
 16  .

where the . represents zero, Alternatively you need to use the IntFFE function to convert 
elements of the Galois field of order 17  into integers. Because you have list of matrices -- ie a list of lists of lists, you do it as follows:

gap> intgens := List(gens, m -> List(m, r-> List(r, IntFFE))); 
[ [ [ 3, 0 ], [ 0, 6 ] ], [ [ 16, 1 ], [ 16, 0 ] ] ]

I don't know if this is appropriate format for matlab, but if  it is, you can write it to a file by

gap> PrintTo( "sl17gens", intgens);


If you have further problems, I'd suggest asking on the GAP support mailing list, rather than the forum. 
We will report back here any advice of wider interest

	Yours

		Steve Linton




> ##    How can we store this output of matrices in a file and make use of
> them in Matlab program ??
> like storing the matrices now generated in a 3d-array and then later using
> them in Matlab.
> 
> Thank you again for helping me out.
> 
> Regards,
> Piyush Garg.
> 
> 
> On Mon, Jun 6, 2011 at 7:51 PM, Stephen Linton <sal at cs.st-andrews.ac.uk>wrote:
> 
>> Dear GAP Forum,
>> 
>> As Peter has already reported, if you simply want a generating set then
>> GeneratorsOfGroup is the appropriate function.
>> As you will see from the documentation (try ?GeneratorsSmallest for
>> instance), the functions you called: GeneratorsSmallest and
>> MinimalGeneratingSet
>> make much stronger promises about the result, which GAP has to do much more
>> work to achieve.
>> 
>> An intermediate choice is SmallGeneratingSet, which tries heuristically to
>> reduce the size of the set it returns, but makes no firm promises.
>> 
>> The error message from MinimalGeneratingSet is confusing,  but, in fact GAP
>> has no way to compute this function for general groups. In the next release
>> you will get the rather more helpful error message:
>> 
>> Error, `MinimalGeneratingSet' currently assumes that the group must be
>> solvable.
>> In general, try `SmallGeneratingSet' instead, which returns a generating
>> set that is small but not of guarateed smallest cardinality
>> 
>> It is also possible, looking at the error message that you have mixed up g
>> and G. Remember that GAP is case-sensitive.
>> 
>> Yours
>> 
>>       Steve Linton
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