MAGMA Computational Algebra System

Basic Operations

Subsections

• Accessing Group Information

Accessing Group Information

The functions in this group provide access to basic information stored for a group G.

G . i : Grp, RngIntElt -> GrpElt

The i-th defining generator for G, if i>0. If i<0, then the inverse of the -i-th defining generator is returned. G.0 is equivalent to Identity(G).

Generators(G) : Grp -> { GrpFinElt }

A set containing the defining generators for G.

NumberOfGenerators(G) : Grp -> RngIntElt

Ngens(G) : Grp -> RngIntElt

The number of defining generators for G.

Generic(G) : Grp -> Grp

Given a group G in the category GrpPerm or GrpMat, return the generic group containing G, i.e., the largest group in which G is naturally embedded. The precise definition of generic group depends upon the category to which G belongs.

Parent(g) : GrpElt -> Grp

The parent group G for the group element g.

Example Grp_Generators (H50E13)

> K<z> := GF(2, 3);
> G := SuzukiGroup(8);         
> Generic(G);
GL(4, GF(2, 3))
> Ngens(G);
3
> for i in [1..3] do
>    print "generator", i, G.i;                      
>    print "order", Order(G.i), "\r";
> end for;
generator 1 
[  0   0   0   1]
[  0   0   1   0]
[  0   1   0   0]
[  1   0   0   0]
order 2 


generator 2 
[z^2   0   0   0]
[  0 z^6   0   0]
[  0   0   z   0]
[  0   0   0 z^5]
order 7 


generator 3 
[  1   0   0   0]
[z^2   1   0   0]
[  0   z   1   0]
[z^5 z^3 z^2   1]
order 4 

Orbit(G, M, x) : Grp, Any, Any -> Any

Given a finitely generated group G that acts on the parent structure of x through the map (or user defined function) M, compute the orbit of x under G. Thus, for every generator g of G, M(g) must return a function that can be applied to x or any other element in the parent of x.

If the orbit is infinite, this process will eventually run out of memory.

OrbitClosure(G, M, S) : Grp, Any, Any -> Any

Given a finitely generated group G acting on the universe of S through the map or user defined function M, compute the smallest subset T containing S that is G-invariant. Thus, for every generator g of G, M(g) must return a function that can be applied to an arbitrary element in the universe of S.

If the orbit closure is infinite, this process will eventually run out of memory.