GAP Manual: 7.50. Factorization

Factorization( G, g )

Let G be a group with generators g_1, ..., g_n and let g be an element of G. Factorization returns a representation of g as word in the generators of G.

The group record of G must have a component G.abstractGenerators which contains a list of n abstract words h_1, ..., h_n. Otherwise a list of n abstract generators is bound to G.abstractGenerators. The function returns an abstract word h = h_(i_1)^(e_1) * ... * h_(i_m)^(e_m) such that g_(i_1)^(e_1) * ... * g_(i_m)^(e_m) = g.

    gap> s4 := Group( (1,2,3,4), (1,2) );
    Group( (1,2,3,4), (1,2) )
    gap> Factorization( s4, (1,2,3) );
    x1^3*x2*x1*x2
    gap> (1,2,3,4)^3 * (1,2) * (1,2,3,4) * (1,2);
    (1,2,3) 

The default group function GroupOps.Factorization needs a finite group G. It computes the set of elements of G using a Dimino algorithm, together with a representation of these elements as words in the generators of G.