ABSTRACT
In the previous section we saw examples of groups acting on spaces in the “external world”. In this section we consider examples of group actions within the domain of group theory and use them to obtain results on the structure of groups. We have already done this before, for example, when we introduced the Cayley embedding in (23.20) (where a group G acts on itself by left-translations) or the semidirect product of two groups in (24.14) (where a group A acts on a group B as a group of automorphisms). Here we want to study actions of this kind more thoroughly. Let us present some examples of the types of actions we want to consider.
