25.57 Factor Groups of Ag Groups

It is possible to deal with factor groups of ag groups in three different ways. If an ag group G and a normal subgroup N of G is given, you can construct a new polycyclic presentation for F=G/N using FactorGroup. You can apply all functions for ag groups to that new parent group F and even switch between G and F using the homomorphisms returned by NaturalHomomorphism. See FactorGroup for more information on that kind of factor groups.

But if you are only interested in an easy way to test a property or an easy way to calculate a subgroup of a factor group, the first approach might be too slow, as it involves the construction of a new polycyclic presentation for the factor group together with the creation of a new collector for that factor group. In that case you can use CollectorlessFactorGroup in order to construct a new ag group without initializing a new collector but using records faking ag words instead. But now multiplication is still done in G and the words must be canonicalized with respect to N, so that multiplication in this group is rather slow. However if you for instance want to check if a chief factor, which is not part of the AG series, is central this may be faster then constructing a new collector. But generally FactorGroup should be used.

The third possibility works only for Exponents (see Exponents) and SiftedAgWord (see SiftedAgWord). If you want to compute the action of G on a factor M/N then you can pass M/N as factor group argument using M mod N or FactorArg (see FactorArg).

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