ABSTRACT
14.1 We have observed in the introduction to Chapter 10, that ℤ is not just any group! In fact, some of its remarkable properties are due to the fact that it is generated, as an additive group, by one element which is even the neutral element for another operation. Since a group is, in particular, a set, it makes sense to investigate whether a group can be generated by certain subsets, that is if one applies the construction of inverse and “multiplication” (the group is multiplicatively written) in an iterated way, is then every element of the group eventually obtained as such an expression? This problem is related to the existence of certain morphisms fro ℤ to G (Theorem 14.6). Since G is not abelian, ℤ alone is not sufficient to express fully the properties of sets of generators, for this it would be necessary to introduce the free group on some given subset, but we do not treat this here.
