Factor morphisms and centroids of locally nilpotent groups. (Q694220)

The author considers endomorphisms \(\sigma\) of a locally nilpotent group \(G\) satisfying two of the following conditions for all \(g,h\in G\) and ordinal numbers \(\alpha\): (a) \(g^{h\sigma}=g^{\sigma h}\), (b) \(g^\sigma h^\sigma=(gh)^\sigma\) if \([g,h]=1\), (c) as in (b), but each as congruences modulo \(Z_\alpha(G)\). The set \(\Gamma(G)\) of mappings satisfying (a) and (b) is called centroid while \(\Phi(G)\) denotes the set of mappings satisfying (a) and (c), the factor morphisms. The author compares these concepts with each other and with the ring centroid concept. Main Theorem: \(\Phi(G)=\Gamma(G)\) if \(G\) is torsion-free locally nilpotent.