SYMMETRICA-MANUAL -- Generators of permutation groups

Generators of permutation groups

Generators of permutation groups

A system of generators of a permutation group in SYMMETRICA is considered to be a VECTOR, where each entry of this VECTOR is a PERMUTATION object (in VECTOR-form), all of the same degree.

In order to compute a system of generators for some standard actions you can use

INT gen_Cn(a,b)    OP a,b; 
INT gen_Dn(a,b)    OP a,b; 
INT gen_An(a,b)    OP a,b; 
INT gen_An_3(a,b)  OP a,b; 
INT gen_Sn(a,b)    OP a,b; 
INT gen_In(a,b)    OP a,b;

As their names imply one can compute the generators of the natural actions of the cyclic group Cn, the dihedral group Dn, the alternating group An, the symmetric group Sn and the trivial group consisting of the identity only In respectively. In gen_An_3 the set of all cycles of length 3 is computed as the system of generators of the alternating group. In all these cases a is the degree of the permutation group (i.e. the number of elements of the set which the group is acting on). b is the VECTOR of generators. a and b must be different.

There is another routine called

INT gen_arb(a)      OP a;

which serves as an input routine for a, an arbitrary system of generators.

harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001

Generators of permutation groups