[GAP Forum] Construction of a group unsing a multiplication table.

[Stephen Linton]

Dear GAP Forum,

Marc Bogaerts asked about realising a particular group construction in GAP> 

If I read the description correctly, this group is generated by N and i = (-1,1_N) with the following rules

n*i = i*n^t
i^2 = g

One approach that would work to build this in GAP for a pretty good range of groups
would be make N as a finitely-presented group N = <n_1..n_k | R> and then simply add i as an additional generator 
 

G = <i,n_1..n_k | R , n_j*i = i*n_j^t, i^2 = g>

GAP should be able to study this group for a pretty decent range of sizes of N. The starting point is IsomorphismFpGroup.

An alternative approach would be to use the technique described in section 4.12 of the reference manual to implement your 
elements directly.

	Steve


On 1 Jun 2011, at 13:43, mbg nimda wrote:

> Dear Forum Members,
> 
> I start with a group N, e.g. N=SL(2,25). Now for a suitable element g in N
> and a suitable automorphism t I define a multiplication rule mult on the
> cartesian product car:=[-1,1] x N by:
> (-1, n) * (-1,m) = ( 1, g*n^t*m)
> (-1, n) * (1,m)  = (-1, n*m)
> (1, n)* (-1,m) = (-1, n^t*m)
> (1,n)*(1,m) = (1, n*m)
> For smaller groups I use the following:
> 1) First I construct a multiplication table using PermList:
> mtab:=List(car,x->List(car, y->Position(car, mult(x,y))));
> 2) M:=MagmaByMultiplicationTable(mtab);
> 3) G:=AsGroup(M);
> But this obviously doesn't work for larger groups, so instead I use the
> following:
> 
> 
> makegroup:=function(car)
> local Gr,gens,x;
> gens:=[];Gr:=(());
> Gr:=Group(());gens:=[];
> for x in car do
> Display(x);
> x:=List(car, y->Position(car, mult(x,y)));
> x:=PermList(x);
> if x in Gr then continue; fi;
> Add(gens,x); Gr:=Group(gens);
> if Size(Gr)>=Size(N)*2 then break; fi;
> od;
> return(Gr);
> end;
> 
> 
> But in this case the size of N is too  big to even construct one generator.
> Is there any suggestion?
> 
> I dont even have to know the group itself, but only its structure.
> 
> 
> 
> Marc Bogaerts
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