§ Internal versus External semidirect products
Say we have an inner semidirect product. This means we have subgroups such that ,
normal in and . Given such conditions, we can realize and
as a semidirect product, where the action of on is given by conjugation in .
So, concretely, let's think of (as an abstract group) and (as an abstract group)
with acting on (by conjugation inside ). We write the action of on
as . We then have a homomorphism
given by . To check this is well-defined, let's take , with
and . Then we get:
So, really is a homomorphism from the external description (given in terms of the conjugation)
and the internal description (given in terms of the multiplication).
We can also go the other direction, to start from the internal definition and get to the conjugation.
Let and . We want to multiply them, and show that the multiplication
gives us some other term of the form :
So, the collection of elements of the form in is closed. We can check that the other properties
hold as well.