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This is new variant of For $h \in G$ and $\phi \in Aut(G)$ is $\phi^n(h)$ periodic in any finite quotient? thanks to Alon's comment. Since that is the case I'm interested really in, I figured it's worth posting the question again with the condition that $G$ be finitely generated:

Let $G$ be a finitely generated infinite group, and $\phi$ an automorphism of it. Let $N$ be a normal subgroup of $G$ such that $G/N$ is finite. Is it true that for any $h$ in $G$, $\phi^n(h)N$ (as a sequence of elements in $G/N$ for $n=1,2,3,...$) is periodic?

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up vote 2 down vote accepted

Yes. $\phi$ descends to an isomorphism $\phi : G/N \to G/\phi(N)$. Let $H = G/N$. Since $G$ is finitely-generated, there are finitely many homomorphisms $G \to H$, so it follows that there are only finitely many normal subgroups of the form $\phi^n(N), n \in \mathbb{Z}$. The rest is clear.

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If $G$ is finitely generated, then there are only finitely many distinct epimorphisms $G \rightarrow G/N$, so there are only finitely many kernels of such homomorphisms, and hence only finitely many images of $N$ under powers of $\phi$. Let $K$ be the intersection of these images of $N$. Then $K \le N < G$, and $K$ is fixed by $\phi$, and $\phi$ induces an automorphism of $G/K$ of finite order. The periodicity of $\phi^n(h)N$ follows.

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