Abstract
Let $\phi$ be an automorphism of a group $G$. Under various finiteness or solubility hypotheses, for example under polycyclicity, the commutator subgroup $[G, \phi]$ has finite index in $G$ if the fixed-point set $C_{G}(\phi)$ of $\phi$ in $G$ is finite, but not conversely, even for polycyclic groups $G$. Here we consider a stronger, yet natural, notion of what it means for $[G, \phi]$ to have 'finite index' in $G$ and show that in many situations, including $G$ polycyclic, it is equivalent to $C_{G}(\phi)$ being finite.
Citation
B. A. F. Wehrfritz. "On the fixed-point set of an automorphism of a group." Publ. Mat. 57 (1) 139 - 153, 2013.
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