Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ which are not in the inner set, given an outer automorphism $\phi(x)$, we can create a new group which has the presentation $(Gen \cup \{\psi\}|Rel \cup \{ \psi x \psi^{-1} = \phi(x) | x \in Gen\})$. I've been referring to this notion by saying we want to internalize the outer automorphism $\phi(x)$. Some questions I've had so far are as follows:

What properties of the original group are preserved by this process?

Do most groups become trivial or infinite? (So far its looking like the latter)

I find this to be really difficult so I really hope somebody here will have some answers for these questions, or at least a place to look or a reason to expect this to be very difficult to answer. So far I've done this for some cyclic groups and they become evidently massive, though I can't prove that because I don't know how, as I just started reading about finite presentations and this sort of thing recently.

1more comment